A Course on Finite Groups Universitext
For other titles in this series, go to www.springer.com/series/223 H.E. Rose
A Course on Finite Groups H.E. Rose Department of Mathematics University of Bristol University Walk Bristol BS8 1TW, UK [email protected]
Editorial board: Sheldon Axler, San Francisco State University Vincenzo Capasso, Universitá degli Studi di Milano Carles Casacuberta, Universitat de Barcelona Angus Macintyre, Queen Mary, University of London Kenneth Ribet, University of California, Berkeley Claude Sabbah, CNRS, École Polytechnique Endre Süli, University of Oxford Wojbor Woyczynski, Case Western Reserve University
ISBN 978-1-84882-888-9 e-ISBN 978-1-84882-889-6 DOI 10.1007/978-1-84882-889-6 Springer London Dordrecht Heidelberg New York
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Springer is part of Springer Science+Business Media (www.springer.com) Preface
This book is an introduction to the remarkable range and variety of finite group the- ory for undergraduate and beginning graduate mathematicians, and all others with an interest in the subject. My original plan was to develop the theory to the point where I could present the proofs and supporting material for some of the main re- sults in the subject. These were to include the theorems of Lagrange, Sylow, Burn- side (Normal Complement), Jordan–Hölder, Hall and Schur–Zassenhaus amongst others, and to provide an introduction to character theory developed to the point where Burnside’s pr qs -theorem could be derived and Frobenius kernels and com- plements could be introduced. I have come to realise that this would have resulted in a rather long book and so some material would have to go. It was at this point that modern technology came to my aid. Solutions to the problems were also to be included, but these would have taken at least 90 rather dense pages and an appendix to this book was perhaps not the best place for this material. A number of textbooks now put solutions on a web site attached to the book which is maintained jointly by the author and the publisher. Extending this idea has allowed me to fulfil my original intentions and keep the printed text to manageable proportions. So the web site now attached to this book, which can be found by going to www.springer.com and following the product links, includes not only the Solution Appendix but also extra sections to many of the chapters and two extra web chapters. These items are listed on the contents pages, and present work that is not basic to a chapter’s topic being either slightly more specialised or slightly more challenging. Also, perhaps unfortunately, all work on character theory and applications (Chapters 13 and 14) is now on the web. As this book goes to press, about half of this web material is written and ‘latexed’, it is hoped that the remaining half will be available when the book is published or soon after. Of course, more web items could be added later. I attended Muchio Suzuki’s graduate group theory lectures given at the University of Illinois in 1974 and 1975, and so in tribute to him and the insight he gave into modern finite group theory I have ended the extended text with a discussion of his simple groups Sz(2n) as an application of the Frobenius theory.
v vi Preface
Prerequisites
This book begins with the definition of a group, and Appendices A and B give a brief résumé of the background material from Set Theory and Number Theory that is required. So in one sense, the book needs no prerequisites, only the ability to ‘think-straight’ and a desire to learn the subject. On the other hand, it would help if the reader had undertaken the following. (a) We are assuming that the reader is familiar with the material of a basic abstract algebra course, and so he or she has seen at least a few examples of groups and fields, associative and commutative operations, et cetera, and also has had some experience working in an abstract setting. (b) We are also assuming that the reader is familiar with the basics of linear al- gebra including facts about vector spaces, matrices and determinants, and the definitions of inner and Hermitian forms. We also use the elementary opera- tions, similarity and rational canonical forms, and related topics. Most standard one-semester linear algebra textbooks provide more than is required. (c) It would also help if the reader had undertaken a first course on analysis which included the basic set operations, elementary properties of the standard num- ber systems: integers Z, rational numbers Q, real numbers R, and the complex numbers C, and the standard set-theoretic methods summarised in Appendix A. (d) Lastly, some familiarity with elementary number theory would be an asset, Ap- pendix B summarises most that is required. The Euclidean Algorithm is used widely in this book, as are the basic congruence properties.
Plan of the Book
The author of an introductory group theory text has a problem: the theory is self- contained and coherent, many topics are interconnected, and several are needed more or less from the start. On the other hand, the material in a book has perforce to be presented linearly starting at Page 1. During the planning and writing of this book, I have assumed that most readers will not read it sequentially from cover to cover, but will occasionally ‘dot-about’. Hence I have allowed some ‘forward refer- ence’, mostly for examples. The essential topics that the reader should ‘get to grips with’ first include the basic facts about groups and subgroups, homomorphisms and isomorphisms, direct products and solubility. Also some aspects of the theory of actions—conjugacy, the centraliser and the normaliser—are not far behind. Of course, as noted above, although the material has to be presented linearly, it need not be read linearly, and there are considerable advantages in presenting the basic facts of a topic— homomorphisms, for example—in one place. One consequence of this fact is that the order of the chapters has some flexibility. So Chapter 7 could be read before Chapters 5 and 6 with only a small amount of back-reference in the examples. Some group-theorists may consider it essential for students to have a good grounding in the Abelian theory before the non-Abelian theory is tackled. Similarly, Chapters 10 Preface vii and 11 can be read in either order with little back-reference required. So a possible non-linear reading of the text is Sections 2.1, 2.3, 2.4 and 4.1—the basic core of the subject, then the rest of Chapter 2, Sections 4.2, 4.3, 7.1 and 11.1 in this order, then the following sections where the reading order might be varied Part or all of Chapters 3 and 5, Sections 7.2, 7.3, and 9.1, and Chapters 6 and 10. Following this the remaining printed sections or possibly some of the web sections could be tackled. In the text, I have sometimes introduced topics ‘early’ and out of their logical order, for example, isomorphisms in Chapter 2, to deal with this point. Also, as a general rule, the ‘easier’ and/or more elementary parts of a topic come near the beginning of the chapter, and so the final sections often contain more specialised and/or challenging material.
Further Reading
The reader would do not harm studying any of the books listed in the bibliogra- phy, we suggest a few concentrating on the more recent titles. For a general further development of the finite theory try: Robinson (1982), Suzuki (1982, 1986), Aschbacher (1986), Kurzweil and Stellmacher (2004), and Isaacs (2008). Also the three volume Huppert and Blackburn (1967, 1982a, 1982b) is very com- prehensive and deals with many topics not found elsewhere. For more specialised topics, the following should be read: Doerk and Hawkes (1992) for soluble groups, Carter (1972), the ATLAS (1985), and Conway and Sloane (1993) for finite simple groups, James and Liebeck (1993), Huppert (1998) and Isaacs (2006) for character theory, and Kaplansky (1969), Fuchs (1970, 1973), and Rotman (1994) for infinite Abelian groups. Of course, some of the older books still have much to offer, these include Burnside (1911, reprinted 2004), Kurosh (1955), Scott (1964) and Rose (1978)—no relation! Although 45 years old, in my opinion, Scott’s book remains one of the best in- troductions to the subject. viii Preface
All errors and omissions that are still present in the text and/or web pages are entirely my fault, please contact me with details at [email protected] General comments, including comments on the correctness and/or clarity of the text, or shorter, clearer or better solutions to the problems (which could be added to the web site), are also welcome.
Acknowledgements I have received a considerable amount of assistance during the writing of this book for which I am extremely grateful. First, from my fam- ily (especially from my wife Rita), from colleagues, both academic and compu- tational (especially Richard Lewis and Peter Burton), at the University of Bristol, and from the staff (both editorial and ‘Latex’ specialist) at Springer Verlag. I have given courses based on preliminary versions of this book many times in Bristol, my students have helped me to clarify many points and I thank them for this. But my main debt of gratitude goes to those who read a final draft of the text and cleared up many inconsistencies and errors on my part. These include the referees appointed by Springer Verlag, John Bowers (formally of the University of Leeds), Robin Chap- man (Universities of Bristol and Exeter), Robert Curtis for Chapter 12 (University of Birmingham), and Ben Fairbairn (University of Birmingham). These last four spent many hours going through the manuscript, and improved it greatly—they are forever in my debt. Bristol Harvey Rose Contents
1 Introduction—The Group Concept ...... 1
2 Elementary Group Properties ...... 11 2.1BasicDefinitions...... 11 2.2Examples...... 17 2.3 Subgroups, Cosets and Lagrange’s Theorem ...... 24 2.4 Normal Subgroups ...... 30 2.5Problems...... 34
3 Group Construction and Representation ...... 41 3.1Permutations...... 42 3.2 Permutation Groups ...... 48 3.3 Matrix Groups ...... 52 3.4GroupPresentation...... 55 3.5Problems...... 59
4 Homomorphisms ...... 67 4.1HomomorphismsandIsomorphisms...... 69 4.2 Isomorphism Theorems ...... 74 4.3 Cyclic Groups ...... 79 4.4 Automorphism Groups ...... 81 4.5Problems...... 84
5 Action and the Orbit–Stabiliser Theorem ...... 91 5.1Actions...... 92 5.2ThreeImportantExamples...... 99 5.3Problems...... 107
6 p-Groups and Sylow Theory ...... 113 6.1 Finite p-Groups ...... 114 6.2 Sylow Theory ...... 119 6.3Applications...... 126 6.4Problems...... 131 ix x Contents
7 Products and Abelian Groups ...... 139 7.1 Direct Products ...... 140 7.2 Finite Abelian Groups ...... 146 7.3 Semi-direct Products ...... 151 7.4Problems...... 159 8 Groups of Order 24, Three Examples ...... 165 8.1 Symmetric Group S4 ...... 165 8.2 Special Linear Group SL2(3) ...... 172 8.3 Exceptional Group E ...... 177 8.4Problems...... 183 9 Series, Jordan–Hölder Theorem and the Extension Problem .....187 9.1 Composition Series and the Jordan–Hölder Theorem ...... 188 9.2ExtensionProblem...... 196 9.3Problems...... 205
10 Nilpotency ...... 209 10.1 Nilpotent Groups ...... 210 10.2 Frattini and Fitting Subgroups ...... 217 10.3Problems...... 223 11 Solubility ...... 229 11.1 Soluble Groups ...... 230 11.2 Hall’s Theorems and Solubility Conditions ...... 236 11.3Problems...... 243 12 Simple Groups of Order Less than 10000 ...... 249 12.1SteinerSystems...... 250 12.2 Linear Groups ...... 254 12.3 Unitary Groups ...... 265 12.4 Mathieu Groups ...... 267 12.5Problems...... 270 Appendices A to E ...... 277 A Set Theory ...... 277 B Number Theory ...... 284 C Data on Groups of Order 24 ...... 289 D Numbers of Groups with Order up to 520 ...... 293 E Representations of L2(q) with Order < 10000 ...... 295 Bibliography ...... 297 Notation Index ...... 301 1—SymbolIndex...... 301 2—Notation for Classes of Groups ...... 303 3—Notation for Individual Groups ...... 304 Index ...... 305 Web Contents
3 Group Construction and Representation ...... 321 3.6 Representations of A5 3.7 Further Problems
4 Homomorphisms ...... 331 4.6 The Transfer 4.7 Group Presentation, Part 2 4.8 Further Problems
5 Action and the Orbit–Stabiliser Theorem ...... 351 5.4 Transitive and Primitive Permutation Groups 5.5 Further Problems
6 p-Groups and Sylow Theory ...... 361 6.5 Applications 2—Burnside’s Normal Complement Theorem and Groups with Cyclic Sylow Subgroups 6.6 Further Problems
7 Products and Abelian Groups ...... 371 7.5 Infinite Abelian Groups—A Brief Introduction
9 Series, Jordan–Hölder Theorem and the Extension Problem .....379 9.4 Schur–Zassenhaus Theorem 9.5 Further Problems
12 Simple Groups of Order Less than 10000 ...... 387 12.6 Simple Groups of Order Less than 1000000, Iwasawa’s Lemma and a Method for generating Steiner Systems for some Mathieu Groups 12.7 Further Problems
13 Representation and Character Theory ...... 401 13.1 Representations and Modules
xi xii Web Contents
13.2 Theorems of Schur and Maschke 13.3 Characters and Orthogonality Relations 13.4 Lifts and Normal Subgroups 13.5 Problems
14 Character Tables and Theorems of Burnside and Frobenius .....433 14.1 Character Tables 14.2 Burnside’s pr qs -Theorem 14.3 Frobenius Groups 14.4 Problems
Solution Appendix—Answers and Solutions, Problems 2 to 12, A and B ...... 461 Problem2 ...... 461 Problem3 ...... 472 Problem4 ...... 481 Problem5 ...... 489 Problem6 ...... 497 Problem7 ...... 508 Problem8 ...... 515 Problem9 ...... 521 Problem10...... 524 Problem11...... 532 Problem12...... 538 ProblemA...... 544 ProblemB...... 545 Chapter 1 Introduction—The Group Concept
Groups are all-pervasive in mathematics, there is hardly a branch of the subject that does not use them in one way or another. They are also widely used in many branches of the physical sciences. In one sense, this is to be expected because groups are quite often formed when an operation like multiplication or composition is ap- plied to a set or system. Groups occur as number systems or collections of matrices, in permutation theory, as the symmetries of geometrical objects or as sets of maps, and in many other guises. Also the theory contains many elegant, dramatic and illu- minating theorems. Group theory has developed sometimes slowly but at other times by great leaps and bounds over the past two centuries. Often ideas and results appeared first im- plicitly before they were explicitly written down. For example, it is thought that Galois, in the 1820s, was the first to write down the axioms of a group, but some forty years earlier Lagrange was working with permutations of the roots of equa- tions and proved a result which led to the famous theorem that now bears his name, the comments on actions given in the Introduction to Chapter 5 also apply here. Galois introduced a number of other basic notions including, for example, simple groups and normal subgroups. In 1850, Cayley showed that every group can be rep- resented as a permutation group, and much of the nineteenth century work dealt with this aspect of the theory. Results started to appear more quickly, Sylow produced his ground-breaking work on p-subgroups in 1872, characters and representation theory were introduced around the turn of the century, and Hall’s extensions of Sy- low’s work appeared in the 1920s and 1930s—to mention only a few of the many major developments. Another surge began around 1950 and led in the early 1980s to the completion of the classification problem for finite simple group, hereafter re- ferred to as CFSG, which must surely rank amongst the greatest achievements in all mathematics. A number of important corollaries have followed from this work, for example, the positive solution of the Restricted Burnside Problem (page 27). The purpose of this book is to introduce the reader to the fine branch of mathe- matics called group theory—there is a ‘great story’ to tell, and we hope that it will encourage you, the reader, to develop an abiding interest in the subject, and a desire to look further and deeper into the theory.
H.E. Rose, A Course on Finite Groups, 1 Universitext, DOI 10.1007/978-1-84882-889-6_1, © Springer-Verlag London Limited 2009 2 1 Introduction—The Group Concept
Group Examples
A group is a mathematical system (or set) with a single operation. We begin by considering two familiar examples; the first is the integers Z. The elements are
...,−2, −1, 0, 1, 2, 3,..., and the operation is standard addition ‘+’. There are a number of basic axioms from which almost all additive properties follow. The first and in some ways the most important is closure, or being well-defined; that is,
if a,b ∈ Z, then a + b ∈ Z.
This is equivalent to stating that + is an operation (Appendix A). Some so-called partial systems have been studied, but all systems considered in this book satisfy an axiom of this type. The next property is associativity; that is,
for all a,b,c ∈ Z we have (a + b) + c = a + (b + c).
When forming this sequence of additions, we obtain the same answer if we first add a to b, and then add the result of this addition to c,orifwefirstaddb to c and then form the sum of a and the result of this last addition. Some algebraic systems lack this property but, in general, non-associative systems have limited uses unless some more complex rule is applied—for example, in Lie algebras—and again we shall not consider such systems in this book. In Z, a natural question to ask is: Does the equation
a + x = b (1.1) have a solution x? In ancient times mathematicians only ‘allowed’ this equation to have a solution if b>a,thatis,ifx is positive. But this is very restrictive, and in the group Z Equation (1.1) is always uniquely soluble, and so we need to introduce the ‘zero’ and ‘negative’ integers. The zero 0 satisfies
a + 0 = 0 + a = a for all a ∈ Z; in the sequel, we use the term neutral element for the entity 0; see the discussion on page 4. Further, we introduce the negative integers by
for all a ∈ Z there exists a unique c ∈ Z that satisfies a + c = 0.
We usually write −a for c (and a−1 for c if we are using a multiplicative notation as is the case for almost all groups discussed in this book), and we call it the inverse of a. It is now easy to show that (1.1) always has a unique solution. The system Z has one extra basic property not shared by all groups: it is commutative,orAbelian. This is given by
for all a,b ∈ Z we have a + b = b + a, 1 Introduction—The Group Concept 3 the result of several additions does not depend on the order of the terms in the sum. To recap, Z has the four basic properties: Closure, associativity, a neutral element and inverses, and it has the extra property of Abelianness. Note also it is a countably infinite system. For our second example, we consider another familiar system—symmetries of an equilateral triangle. Groups of symmetries provide a good range of examples, they are widely used in both mathematical and physical systems, for example, by chemists when they are studying the crystal structure of matter. The group of sym- metries of a triangle has two aspects which are different from those in our first example: It is finite, and it is not Abelian, but as we shall see below it shares the four basic properties with Z. Consider an equilateral triangle with vertices labelled A,B,C. This geometric object has a number of symmetries, that is, transformations (rotations and reflections) that give another copy of the original triangle. We work in the standard Euclidean plane.
A C B α α → −→ −→ → C B B A A C ||| ↓↓↓βββ
A C B α2 α2 → −→ −→ → B C A B C A
The elements of the group are the six rotations and reflections of the triangle illus- trated above, and the operation is composition; that is, do one rotation or reflection, and then do another on the result of the first. We take the basic rotation α to be clockwise about the centre of the triangle by the angle 2π/3,seethetoprowofthe diagram above. Note that three applications of α (that is, α3, a rotation by 2π) maps the triangle to itself identically, and so can be taken as the neutral element. This also shows that the inverse of α is α2; see the bottom row in the diagram. The triangle has three reflections, they are mirror transformations about a line through a vertex and the centre of the corresponding opposite edge. One, labelled β (three times) in the diagram, is about a vertical line through the top vertex of the corresponding triangle and the centre of its base. The other two reflections can be generated as follows. If we first apply α to the top-left triangle and then apply β to the result, we obtain the middle triangle in the bottom row. This gives the second reflection of the top-left triangle, now about a line through the bottom right-hand vertex B and the centre of the opposite edge AC. Note that we would obtain the same re- sult if we first applied β to the top-left triangle, and then applied α2 to the result. We can obtain the third reflection if we repeat this construction but begin by apply- 4 1 Introduction—The Group Concept ing α2 instead of α to the top-left triangle. Incidentally, this shows that the group is not Abelian (that is, not commutative as βα = α2β = αβ). The inverse of a re- flection is itself: Two applications of a reflection gives the neutral element. We call a group element of this type an involution, and we shall see later that these ele- ments play an important role in the theory. It is straightforward to check that this system is closed and has the associativity property (reader, try a few cases). Hence the system contains the six symmetries of the triangle: Neutral element (where a triangle is mapped to itself identically), α, α2, β, αβ, and α2β, and it satisfies the four basic group axioms (closure, associativity, and possession of a neutral element and inverses for all of its elements) as in our first example. It is also finite and not Abelian. Later we shall call this system the dihedral group of the triangle and denote it by D3.
Abstract Groups and Representations
With these and other examples in mind, we define a group as a system with a single operation satisfying the four basic properties (axioms) described above, the for- mal definition is given at the beginning of Chapter 2. Also, Section 2.2 provides a substantial list of examples. The following point is important. In both cases, the examples given above are particular ‘instances’ of the group in question; we call them representations of the group. Referring to the second example, another repre- sentation is given by considering the set of permutations of the set {1, 2, 3} with composition as the operation; see page 19 and Section 3.1. A third representa- tion using 2 × 2 matrices is given in Problem 4.2. Each group example given in this book is a representation of its corresponding abstract group, see the discus- sion in the Introduction to Chapter 3. Right from the start this is a characteristic feature that the reader should note, and we use a corresponding nomenclature. So when discussing a group in the abstract we call it a ‘group’, but when dis- cussing a particular example or representation, say using permutations or matri- ces or some analogous construction, we call it a ‘permutation group’ or a ‘matrix group’ or analogous group. Similarly, the ‘neutral element’ which we always de- note by e is the element in the abstract group satisfying the relevant axiom, and it has many instances or representations in particular groups. In the examples given in Section 2.2,itis0,or1,or−3, or a matrix, or a collection of maps, or the point at infinity on a curve, et cetera. The term ‘neutral element’ is not standard, nor is it ‘new’; for example, it occurs in Cohn (1965), page 50. Some authors use either ‘identity’ or ‘trivial element’; the first only really applies to an ‘oper- ation of multiplicative type’, whilst the second gives the wrong impression—this element is an important one in the group, and certainly not trivial in the usual sense of this word. We have extended this nomenclature to the single element (sub)group e which we call the neutral (sub)group, see Definition 2.12.Alsoan ‘inverse’ can be an additive inverse, or a multiplicative inverse, or an inverse matrix, et cetera. 1 Introduction—The Group Concept 5
Classes of Groups
The class of all groups is a large one. Set-theorists call it a proper class as opposed to a set, but as we are taking the usual naive view of set theory (Appendix A) we shall treat sets and classes synonymously. We shall see that it is convenient to consider subclasses defined by some of the basic properties. For example, groups can be finite or infinite, and Abelian or non-Abelian; these distinctions are fundamental. We shall study further distinctions later, for instance, in Chapter 11 between ‘soluble’ and ‘non-soluble’ groups. Here we divide the class of all groups into four subclasses and, as we shall see, both the theory and the actual groups in each subclass have distinct characteristics.
The first subclass contains the
Finite Abelian Groups In Chapter 7, we show that groups in this class can be characterised completely, and they have a particularly simple form—that is, as ‘products of cyclic groups’. So in one sense they are ‘a bit boring’; but in an ap- plication, if we know apriorithat the group or groups under discussion are in this subclass, then we can be sure that they take this simple form which can have a major influence on the result. A good example occurs in the theory of rational points on elliptic curves discussed on page 22. Amongst our four subclasses, this is the only one for which we have a complete description of all of the groups involved.
The second subclass contains the
Finite Non-Abelian Groups Most of the work in this book deals with groups in this class. For finite groups in general, there is a strong interplay between the ‘group theory’ and the ‘number theory’ of the group in question. In part this is a conse- quence of Lagrange’s Theorem which states that the order (number of elements) of a subgroup H of a group G must divide the order of G; so the prime factorisation of the order of a finite group is an important invariant of the group. One major de- velopment from this is the Sylow theory discussed in Chapter 6 which asserts the existence of subgroups with prime power order. Another important distinction in the theory is between the so-called ‘simple’ and ‘non-simple’ groups; see the definition on page 33. The Jordan–Hölder Theorem states, roughly speaking, that all finite (and some infinite) groups can be ‘built up’ from simple groups using ‘extensions’; this will be discussed in Chapter 9—note that one of the main aims our work is to describe all groups. A theory of extensions has been developed, but a considerable amount of work and many new ideas will be needed before it can be described as finished; see Section 9.2. On the other hand, a complete list of finite simple groups is now known, much of the development work was undertaken between 1955 and 1985 and, as noted above, it forms one of the crowning achievements of twentieth century mathematics. We give a brief introduction to this topic in Chapter 12. Hence considerable progress has been made in the theory of finite non-Abelian groups and this will be discussed in the following chapters and web appendices, but work still needs to be done. 6 1 Introduction—The Group Concept
The third subclass contains the
Infinite Abelian Groups For infinite groups in general, number theory only plays a small role, but questions concerning cardinality can be important. The so-called finitely-generated Abelian groups are similar to those in the first subclass as they can be represented as products of cyclic groups. But many groups in this class are not finitely generated, for example, the rational numbers with addition or the positive reals with multiplication. Apart from a brief survey in Web Section 7.5 we shall not deal with these groups in this book. Much of the work develops ideas from linear algebra, and good introductions to this topic are given in Kaplansky (1969), and Fuchs (1970, 1973).
The final subclass contains the
Infinite Non-Abelian Groups This is perhaps the least well-understood part of the theory. A number of extensions of the finite theory have been studied, but no general classification is known, and it seems unlikely that one will be found in the near future. One approach is to use topology. For example, the reals have a natu- ral (metric) topology, and the interplay between the group theory and the topology of this system can be exploited to gain new insights. Since 1950 a number of long- standing problems have been solved, often showing that these groups are more com- plicated than previously thought; for example, see Problem 6.7. A good introduction is given in Kurosh (1955), also Robinson (1982) discusses a number of aspects of this part of the theory. Infinite groups with some kind of ‘finiteness condition’, such as being ‘finitely generated’ or ‘finitely presented’, have also been widely studied.
Summary of the Book
Below we give a brief summary of the contents of the printed Chapters 2 to 12, Appendices A to E,theWeb Chapters 13 and 14, and the Web Appendices.
Chapter 2—Elementary Group Properties The basic entities—semigroups, groups, subgroups, cosets, normal subgroups and simple groups—are defined, La- grange’s Theorem is derived, and the second section lists a number of standard ex- amples.
Chapter 3—Group Construction and Representation The main construction methods and group representations are discussed. Firstly, permutations are intro- duced, the symmetric and alternating groups are defined, and an elementary proof of the simplicity of An for n>4 is given. Secondly, matrix groups are briefly consid- ered, and lastly group presentations are introduced (this topic is completed in Web Section 4.7 once the First Isomorphism Theorem has been proved). Web Sec- tion 3.6 discusses some of the various representations of the alternating group A5 to illustrate the fact that groups can have a wide range of representations. 1 Introduction—The Group Concept 7
Chapter 4—Homomorphisms The natural maps, called Homomorphisms (and Isomorphisms when the map is a bijection), and factor groups are introduced, and the four fundamental Isomorphism and Correspondence Theorems are derived. Cyclic groups and the basic properties of the automorphism group of a group are described. There are two Web Sections 4.6 and 4.7. The first introduces the ‘transfer’ which provides a useful example of a ‘real’ homomorphism, and the sec- ond completes the work on group presentations begun in Chapter 3.
Chapter 5—Action and the Orbit-stabiliser Theorem This chapter, the last giv- ing the basic material, introduces ‘actions’ which bring together a number of useful constructions, and gives a proof of the Orbit-stabiliser Theorem. It also describes three important particular actions: the coset action, the conjugate element action leading to centralisers and the Class Equations, and the conjugate subgroup action leading to normalisers and the N/C-theorem. Web Section 5.5 extends the work on permutation theory begun in Chapter 3, and includes a discussion of ‘transitive’ and ‘primitive’ permutation groups, and Iwasawa’s simplicity lemma.
Chapter 6—p-Groups and Sylow Theory The basic theory of p-groups (where all elements have order a power of p) is discussed, and the five Sylow theorems are derived—these results form one of the most important aspects of the finite the- ory. There are then two sections of applications, the first gives (a) some facts about groups whose orders have a small number of factors, (b) proves the so-called Frattini Argument, and (c) introduces nilpotent groups. The second is Web Section 6.5 which gives some more substantial applications including a proof of Burnside’s Nor- mal Complement Theorem and a discussion of groups all of whose Sylow subgroups are cyclic.
Chapter 7—Products and Abelian Groups Direct products are introduced, and two proofs of the Fundamental Theorem of Abelian Groups are presented; see page 5. The third section discusses ‘semi-direct products’, a variant of the direct product construction, and the groups of order 12 are described (they can all be treated as semi-direct products). Some basic facts, but no proofs, concerning in- finite Abelian groups are given in Web Section 7.5. Except for some problems, this is the only point where specifically infinite groups are considered in any detail.
Chapter 8—Groups of Order 24, Three Examples No new theory is presented in this chapter, but three groups of order 24 are discussed in some detail. The work constructs their subgroups including those of Frattini and Fitting, the subgroup lat- tice, series and some of their representations. Appendix C, see pages 289 to 292, gives data on the remaining twelve groups of order 24. The purpose of this chapter is to challenge the reader to think more about the objects he or she is studying, and to ask questions. For example: can the centre of a group equal its derived subgroup or its Frattini subgroup? This chapter is also intended to motivate the remaining topics, that is series, simple groups and (on the web) representation theory. 8 1 Introduction—The Group Concept
Chapters 9—Series, Jordan–Hölder Theorem and the Extension Problem This is the first of three shorter chapters dealing with series and the normal sub- group structure of groups. In the first of these, we prove the theorem of Jordan and Hölder on composition series—this demonstrates the importance of simple groups to the theory. Secondly, we present a brief introduction to extension theory—that is the construction of complex groups using some of their subgroups as components, and we discuss one substantial example.
Chapter 10—Nilpotency Nilpotent groups lie between Abelian and soluble groups, and this second shorter chapter continues the work on these groups be- gun earlier. There is a surprising number of equivalent definitions which shows the importance of the notion. The second section discusses two ‘special’ subgroups of a group—the Frattini and Fitting subgroups; they have some remarkable properties (including being nilpotent), and extensions of the latter have proved useful in the completion of CFSG.
Chapter 11—Solubility After a brief historical introduction the last of the shorter chapters introduces the basic facts about soluble groups, and discusses a number of equivalent conditions. The most important is due to Philip Hall and extends the Sylow theory in the soluble case.
Chapter 12—Simple Groups of Order Less than 10000 This is another ‘de- scriptive’ chapter giving an account of simple groups with order less than 10000. We introduce Steiner systems—their automorphisms provide a new way to con- struct groups, prove the simplicity of the linear (matrix) groups Ln(q), and discuss one ‘classical’ (U3(3),aunitary group) and one ‘sporadic’ (M11, the first Mathieu group) group in detail. Some numerical data is also given but many proofs are omit- ted. Appendix E, see page 295, gives data on the groups L2(q), and an appendix at Web Section 12.6 provides more information about Steiner systems for Mathieu groups, and data on simple groups of order less than 106.
Appendix A—Set Theory and Appendix B—Number Theory These two ap- pendices give the basic definitions and results for the work on sets and number theory which underlie all the material in the book.
Appendices C, D and E These appendices provide data on several aspects of the theory. The first, C, lists properties of groups of order 24 (this is an appendix to Chapter 8), D details the number of groups with order up to 520, and E provides some representations of the linear groups L2(q) (this is an appendix to Chapter 12).
Web Chapter 13—Representation and Characters A brief introduction to representation and character theory is presented sufficient for the applications given in Web Chapter 14. This theory includes the basic definitions, Schur’s Lemma and Maschke’s theorem, the orthogonality relations, and ‘lifts’, et cetera. This chap- ter is entirely theoretical except for the examples. 1 Introduction—The Group Concept 9
Web Chapter 14—Character Tables, and Theorems of Burnside and Frobe- nius We give three applications of the work in Web Chapter 13 which have strong connections with the earlier material. First, we construct some character ta- bles including those for most of the groups of order 12 or less, and some others including several of order 24 discussed in Chapter 8. These character tables provide a surprising amount of information concerning the groups in question. Second, we prove Burnside’s pr qs -theorem (this completes the proof of Hall’s Theorem given in Chapter 11). The third section introduces the Frobenius ‘kernel’ and ‘comple- ment’, gives a proof of Frobenius’s Theorem concerning these notions, and finally it provides some applications of this theorem including a discussion of Suzuki groups.
Web Solution Appendix This includes answers, hints on solutions, and in some cases full solutions, for all of the problems given in Chapters 2 to 12, and Appendices A and B.
Developing the theory and proving results are of course important, but two other aspects are also important.
Problems Each chapter ends with a sequence of problems for the reader to try of varying difficulty partly as indicated with a star suggesting a greater challenge. Some readers may find it difficult to decide which problems start with and which are the most important, so some of these have been marked with the symbol . These are all fairly straightforward, theoretical, and contain minor results that are used in the main part of the text. Other problems ask for examples to be constructed, these have no indication mark but should also be tackled early. As noted above, hints, sketch solutions, or in some cases detailed treatments of problems, are given in Web Solution Appendix on the web site attached to this book.
‘Actual groups’ We are studying groups, and so it seems essential to us that the reader ‘sees’ and ‘experiences’ as many ‘actual’ or ‘concrete’ groups as possible. This will, we hope, illuminate the theory and so induce a greater understanding in general. Some parts of the text and a number of the problems are given over to this aspect including the whole of Chapter 8.
Computers in Group Theory
During the past thirty years, and more so recently, computers have become an in- creasingly useful tool in pure mathematics, as well as in most other branches of mathematics, many branches of the physical sciences, and beyond. In group theory, they are particularly useful for doing matrix and permutation calculations, and for producing examples. But they can also be used for more sophisticated constructions, for example, looking for subgroups of a group or constructing homomorphisms. A number of computer algebra packages, some ‘free’ and some commercially avail- able, have been developed over the past decade or so, and the reader is encouraged 10 1 Introduction—The Group Concept to make use of at least one of these while reading this book. Also a number of the problems are best tackled using one of these packages. While writing this book, we have made extensive use of the computer algebra package called GAP—Groups, Algorithms, and Programming. This package has many authors based in Aachen in Germany, St. Andrews in Scotland, and at many other sites; we would like to take this opportunity to compliment these authors on the excellence of their product. It is available free from the St. Andrews web site at
http://www-gap.dcs.st-and.ac.uk/~gap
We have also made some use of the commercially available package called MAGMA which incorporates many aspects of the GAP program. One point should be borne in mind whilst working with any of these packages, and it is one that we emphasise several times in this book. In a particular calcula- tion, the program can only deal with a specified representation of the group under discussion, say as a permutation group or as a matrix group. The package GAP is particularly good when working with permutation groups, but it also deals well with matrix groups defined over a specific field and with presentations. Chapter 2 Elementary Group Properties
In this chapter, we introduce our main objects of study—groups. A general overview including some historical comments was given in Chapter 1. More detail on the his- tory of the theory can be found in Wussing (1984), van der Waerden (1985), and at www-gap.dcs.st-andrews.ac.uk/~history/. Here we give the basic definitions and an extensive list of examples, introduce subgroups and cosets, normal subgroups and simple groups, and prove the first major result in the theory—Lagrange’s The- orem.
2.1 Basic Definitions
We begin by defining the group concept. Maps between groups will be discussed in Chapter 4. As a preliminary to this we introduce semigroups as follows.
Definition 2.1 A semigroup is a non-empty set X ={...,x,y,z,...} together with a binary operation (page 281) which satisfies the following two conditions (ax- ioms): (i) it is closed,orwell-defined: for all x,y ∈ X, we can perform the operation x y and
x y ∈ X,
(ii) it is associative: for all x,y,z ∈ X,
x (y z) = (x y) z.
Note that (i) is implied by the definition of the operation ; see the comments below Definition 2.2. H.E. Rose, A Course on Finite Groups, 11 Universitext, DOI 10.1007/978-1-84882-889-6_2, © Springer-Verlag London Limited 2009 12 2 Elementary Group Properties
Examples The following sets with operations are semigroups. (a) The positive integers with addition. (b) The set of all one-variable functions with domain and codomain R, and with the operation of composition of functions.
There is an extensive theory of semigroups which is of particular interest in some branches of analysis and combinatorics. Also a number of similar systems that are not quite groups have been studied, for instance, the operation may be only partially defined, or there may be a neutral element but no inverses, et cetera. We shall not consider these systems; Bruck (1966) provides a good introduction.
Definition 2.2 A group (G, ) is a semigroup which satisfies the following extra conditions (axioms): (iii) G contains a unique element e which satisfies, for all g ∈ G,
e g = g e = g,
(iv) for each g ∈ G, there exists a unique g ∈ G satisfying
g g = g g = e.
The element e is called the neutral element of the group G, see page 4. Some au- thors use the term identity for e, and if the operation ( ) is written additively (+) then it is called the zero and denoted by 0. There are a number of redundancies in this definition—in particular, in axioms (i), (iii) and (iv). Strictly speaking, (i) is un- necessary as it is implied by the fact that is an operation; see Appendix A.Butwe have left it in to remind the reader that closure is vitally important—this property must be checked whenever it is required to show that a particular set and product form a group. For (iii) and (iv), see Theorem 2.5.
Definition 2.3 A group (G, ) is called Abelian, or occasionally commutative,if its operation is commutative: For all g,h ∈ G
g h = h g.
The term ‘Abelian’ commemorates the Norwegian mathematician Niels Abel who died at the age of 27 in 1829. He was working on solutions to polynomial equations, and needed to apply a condition similar to the one above; see the Introduction to Chapter 11 and van der Waerden (1985), page 88.
Examples We give four here, and an extensive list in the next section. (a) The set {1, −1} with the operation of standard multiplication forms a finite Abelian group which we denote by T1 (one copy of ‘two’). The neutral element is 1, and each element is self-inverse. (b) The set of permutations of a set {1, 2, 3}. The elements are the six permutations of this set, and the operation is composition: Do the first permutation, then do the 2.1 Basic Definitions 13
second permutation on the result of the first. For example, if the first permutation maps 1 → 1, 2 → 3 and 3 → 2, and the second maps 1 → 3, 2 → 2 and 3 → 1, then their composition maps 1 → 3, 2 → 1 and 3 → 2. The neutral element is the permutation that moves no symbols, and the inverse of a permutation is its reverse (Section 3.1). This system forms a finite non-Abelian group which we denote by S3 and call the symmetric group of the set {1, 2, 3}. Reader, why is this group not Abelian? (c) The positive real numbers R+ with multiplication form an infinite Abelian group. The neutral element is 1, and the inverse of x is 1/x. (d) The set of all non-singular 2 × 2 matrices having rational number entries with the operation of matrix multiplication is an example of an infinite non-Abelian group, it is denoted by GL2(Q) and called the 2×2 general linear group over Q. The neutral element is I2, the 2-dimensional identity matrix, and inverses exist by definition.
The symbols (G, ), G, H , J ,orK, sometimes with primes or subscripts, will always denote groups. We use lower case Roman letters a, b, c, d, g, h, j, k, and l, again sometimes with primes or suffixes, to stand for group elements, and we use x, y and z for set elements or occasionally for group elements following the usual mathematical convention that these letters denote entities which satisfy a proposition or equation. The words ‘operation’, ‘multiplication’ and ‘product’ are used more or less synonymously: If g,h ∈ G we say that we apply the operation to form the product g h, or we multiply g by h to obtain g h. The underlying set of a group G is the set of elements of G stripped of its oper- ation; where there is no confusion, this will also be denoted by G. Also, we some- times say that a group G is generated by a set X,orX is a generating set for G, where X is a subset of the underlying set of G, and we write G =X. This means that the collection of all products of powers (both positive and negative) of elements of X coincides with G. For example, the set {1} is a generating set for Z, that is, Z =1 because every integer can be expressed as a sum of 1s or −1s. Note that a group may have many different generating sets, and it always has at least one be- cause the underlying set of G clearly acts as a generating set for G. This notion is defined formally in Definition 2.16. We also write e for the (unique) group con- taining the single element e (with operation e e = e), we call it the neutral group. Some authors use the term ‘trivial group’ for e; it is an important component of a group and certainly not ‘trivial’ using the normal meaning of this word, hence we shall not use this term; see also the comments on page 4. We noted above that Definition 2.2 can be weakened considerably without af- fecting our objects of study. Consider
Definition 2.2 A group (G, ) is a semigroup, see Definition 2.1, which satisfies the following two conditions: (iii) there is an element f ∈ G with the property: f g = g, for all g ∈ G; (iv) for each g ∈ G, and with f as in (iii), there exists h ∈ G satisfying h g = f . 14 2 Elementary Group Properties
These conditions imply that G has at least one left neutral element f , and each g ∈ G has at least one left inverse h relative to f . Definition 2.2 is equivalent to Definition 2.2; see also Problem 2.4. Note that this equivalence is useful, for when checking if the group axioms hold for a particular set and map, once closure and associativity have been established (Axioms (i) and (ii)), it is not necessary to prove that either the neutral element or the inverse operation is unique, or two-sided, because these properties follow by Theorem 2.5 below. Also, if we find an inverse of an element g, then we can be sure that it is the unique inverse of g, again by Theorem 2.5. We begin with the following result: In all groups, the only element which equals its square is the neutral element (in algebra generally, such elements are called idem- potents).
Lemma 2.4 Let (G, ) be a semigroup satisfying the conditions of Definition 2.2. If a ∈ (G, ) and a a = a, then a = f where f is given by (iii).
Proof 1 By (iv),ifa ∈ G we can find b ∈ G satisfying b a = f , so by (iii)
a = f a = (b a) a = b (a a) = b a = f,
by associativity, the hypothesis and (iv) again.
Theorem 2.5 A semigroup (G, ) satisfying Conditions (iii) and (iv) in Defini- tion 2.2 forms a group as given by Definition 2.2.
Proof We need to show that f , and the inverses, apply both on the left and on the right, and are unique; that is, f as the neutral element, and h as the inverse of g. First, we show that if a ∈ (G, ) and b a = f , then a b = f (a left inverse is also a right inverse). We have
b (a b) = (b a) b = f b = b.
by (ii), (iv), and (iii). Hence, by (ii) again
(a b) (a b) = a (b (a b)) = a b.
By Lemma 2.4, this shows that a b = f ; the first part follows. Secondly f is a right identity. We have, using the above subresult and (ii),
a f = a (b a) = (a b) a = f a = a
by (iii). Thirdly, we show that b is unique (that is, inverses are unique). For suppose c a = f , then we have by the above and (ii)
c = c f = c (a b) = (c a) b = f b = b,
1To emphasise their importance, and to aid clarity, all proofs are typeset indented. 2.1 Basic Definitions 15
by hypothesis and (iii) applied to b. Lastly, the neutral element. For if e also satisfies (iii), that is e a = a for a ∈ G, then substituting e for a we obtain e e = e, and so, by Lemma 2.4, e = f . This completes the proof.
From now on, we adopt the following conventions. We write ab for a b, e for the neutral element, and G for (G, ) when it is clear which operation is being used. Also, the inverse of g given by (iv) in Definition 2.2 will be written in the standard notation g−1 (and −g if we are using addition). We normally drop brackets and write xyz for either x(yz),or(xy)z. In some cases, we do not delete the brackets if this aids clarity. The next three results apply to all groups, and they will often be used in the sequel usually without being specifically identified. Note that no restrictions apply, a rare occurrence in the theory!
Theorem 2.6 (Cancellation) Suppose a,b,x,y ∈ G. If ax = bx, or if ya = yb, then a = b.
Proof From ax = bx we obtain, by Definition 2.2 and associativity, a = ae = a xx−1 = (ax)x−1 = (bx)x−1 = b xx−1 = b.
A similar argument applies in the second case.
Theorem 2.7 Suppose a and b are elements of a group G. (i) (ab)−1 = b−1a−1. (ii) (a−1)−1 = a. (iii) If G is finite, then a−1 equals some positive power of a. (iv) If a commutes with b, then a−1 also commutes with b.
Proof (i) As (b−1a−1)(ab) = b−1(a−1a)b = b−1b = e and, by Theorem 2.5, inverses are unique and two-sided, it follows that b−1a−1 is the inverse of ab. A similar argument applies for (ii). (iii) If G has n elements and a = e, then an integer m must exist satisfying 1 ··· −1 = −1 ··· −1 Using induction we can extend (i) to prove that (a1 an) an a1 . Theorem 2.8 If we treat the group G as a set (that is, we consider the underlying set of G) then, for all fixed a ∈ G, G ={ag : g ∈ G}={g−1 : g ∈ G}. Proof Use Theorems 2.6 and 2.7; see Problem 2.1. 16 2 Elementary Group Properties Most elementary exponent properties apply to groups. Note that as the group in question may not be Abelian some properties do not always hold. For example, (ab)2 may, or may not, equal a2b2. For all elements a in a group, if n ≥ 0 we write + a0 = e and an 1 = ana, that is an = aa ···a(ncopies of a). Also, again if n ≥ 0 we write a−n in place of (a−1)n. By Theorem 2.7,thisalso equals (an)−1; reader, why? Theorem 2.9 Suppose a is a group element, and r, s ∈ Z. (i) ar+s = ar as . (ii) (ar )s = ars. Proof (i) By induction on s. Suppose s is non-negative. We have ar+0 = ar = ar e = ar a0 and, using the inductive hypothesis in the third equation, ar+(s+1) = a(r+s)+1 = ar+sa = ar asa = ar as+1. Now apply induction. Using this we have ar−sas = a(r−s)+s = ar , hence ar−s = ar (as)−1 = ar a−s and (i) follows for negative s. (ii) Assume first that s is non-negative. As above we use induction on s, we have (ar )0 = e = a0r and + ar (s 1) = ar sar = arsar = ar(s+1), and this case follows by the inductive hypothesis and (i). The reader should do the remaining case using a similar method to that given in the last part of the proof of (i). We shall see below that an important invariant of a group is the number of ele- ments in its underlying set, we define this as follows. Definition 2.10 (i) Two groups G and H are called equal, G = H , if and only if their underlying sets are equal (page 277), and they have the same operation. (ii) The order of a group G is the number (or cardinality) of elements in the underlying set of G, this is denoted by o(G). Some comments on cardinality are given in Appendix A. One or two authors reserve the word ‘order’ for groups and use the word ‘size’ for sets, we shall use ‘order’ for both. Note that o(G) can be finite or infinite, and this distinction is im- portant; see page 5. If the order of G is finite, then the usual number-theoretic rules apply and, as we shall show later, they have a powerful controlling influence on the structure of G.Ifo(G) is infinite, then different considerations apply and care is needed when interpreting results. 2.2 Examples 17 Isomorphism—A Preliminary Note Several groups can appear to be distinct but are, in fact, identical from the group- theoretical point of view. If we have two groups G1 and G2 with a bijection θ be- tween their underlying sets which preserves or transforms all group-theoretic prop- erties of G1 to G2, and vice versa, then we say they are isomorphic, and θ is an isomorphism between them, symbolically this is written G1 G2. The main prop- erty is (ab)θ = aθ · bθ, (2.1) for all a,b ∈ G1; see page 68. We shall give a formal definition of this concept at the beginning of Chapter 4, but it will be convenient to be able to use this notion from now on. As an illustration, we give two examples here. If G = H ,seeDef- inition 2.10(i), then the identity map (page 281) clearly acts as an isomorphism. Secondly, the real numbers with addition R, and the positive reals with multiplica- tion R+, both form groups. They are isomorphic, and one isomorphism θ defined by xθ = 2x, for x ∈ R, : R → R+ demonstrates this fact. The map θ is a bijection (with inverse log2) and it transfers all group-theoretic properties of the first group to the second, and vice versa via (2.1). For instance, the neutral element 0 of R is mapped to 20 = 1, the neutral element of R+, and if a,b ∈ R then (a + b)θ = 2a+b = 2a2b = aθbθ which verifies (2.1) in this case. Isomorphism Class Consider the statement: “There are only two groups of order 6” (Problem 2.20). This is not correct as it stands because there are infinitely many distinct groups of order 6, but many are isomorphic to one another. So to be more precise, we should say that “there are exactly two isomorphism classes of groups of order 6”. If we take the group with elements {0, 1, 2, 3, 4, 5} and operation addition modulo 6 (Z/6Z,the cyclic group of order 6), and D3 (page 3) as our ‘standard’ groups of order 6, then it is true that all groups of order 6 are isomorphic to one of these two groups. When discussing groups of a fixed size we shall often use this short-hand. 2.2 Examples Groups are found throughout mathematics, there is hardly a branch of the subject where they do not occur, they are also widely used in many branches of the physi- cal sciences. We give here an extensive list of examples to illustrate the range and applicability of the group concept. No proofs will be given, in most cases it is not 18 2 Elementary Group Properties difficult to check that the group axioms are satisfied. Note that the notation for indi- vidual groups given in this section will be used throughout the book, see pages 303 and 304. Number Systems Our first examples are the standard number systems. The integers Z, the rational numbers Q, the real numbers R, and the complex numbers C, with the operation of standard addition in each case, all form infinite Abelian groups. The non-zero ra- tional numbers Q∗, non-zero real numbers R∗, and non-zero complex numbers C∗, with the operation of multiplication in each case, also form infinite Abelian groups distinct from the above. In general, for a ring or field F we let F ∗ denote the mul- tiplicative group of the non-zero elements of F . Further, the positive rationals Q+ with multiplication form a group (which is a subgroup of Q∗; see Section 2.3), with a similar construction for R+. Note that neither the non-zero integers with multipli- cation nor the positive integers with multiplication form groups as inverses do not exist. Modular Arithmetic Our second collection of examples are finite groups from number theory. If m>0, the congruence a ≡ b(mod m) stands for: a and b have the same remainder after division by m (in symbols, m | b − a). This notation was first introduced by C.F. Gauss in 1801 in his famous number theory text called ‘Disquisitiones arithmeticae’. Let Z/mZ denote the set {0, 1,...,m− 1}.Ifa,b ∈ Z/mZ, the operation +m is given by a +m b = a + b, if a + b a +m b = a + b − m, if a + b ≥ m, (so a +m b ≡ a + b(mod m), this is called addition modulo m). The set Z/mZ with the operation +m is an Abelian group of order m, the notation Z/mZ which relates to cosets and factor groups will be explained in Chapter 4. This implies that at least one group of order m exists for each positive integer m; in some cases this is essentially the only group of this order (that is, up to isomorphism); for example, when m = 13 or 15, see Appendix D. If m is a prime number p, then (Z/pZ)∗ = (Z/pZ)\0 with multiplication mod- ulo p, defined similarly to addition modulo p, forms another finite Abelian group. Inverses exist by the Euclidean Algorithm (Theorem B.2 in Appendix B). Also note that the group T1 ={−1, 1} (page 42) is isomorphic both to the group Z/2Z, and to the group (Z/3Z)∗. 2.2 Examples 19 Product Groups Given groups G1,...,Gn, we can form a new group by taking all (ordered) n-tuples of the form (g1,...,gn), where gi ∈ Gi for i = 1,...,n, as the new elements, and defining the new operation component-wise using the operations of each Gi in turn. In many cases, several different operations can be defined; see Chapter 7. For exam- ple, suppose n = 2 and G1 = G2 = T1. The elements of the product group are (1, 1), (1, −1), (−1, 1), (−1, −1), and the operation is given by (a, b)(c, d) = (ac,bd). This group is called the 4-group and is denoted by T2, it is a product of two copies of T1; in Chapter 7, we use the standard notation C2 × C2 for this group. Some authors use K (for Klein group) or V (for ‘Viergruppe’ the German word for ‘4-group’ or ‘fours group’) for this group. Note that the square of every element in T2 is the neutral element (1, 1), and it is an example of an Elementary Abelian Group as defined in Problem 4.18. Matrix Groups Matrix groups form one of the most important collections in the theory. Let F be a field (for instance, the rational numbers Q) and let m ≥ 1. The set of non-singular m × m matrices with entries from F and operation matrix multiplication forms a non-Abelian group (if m>1) called the m × m general linear group over F ,itis denoted by GLm(F ). The group axioms can be shown to hold using some elemen- tary matrix algebra; the matrices are non-singular, and so inverses exist by definition. See Section 3.3 for further details. As we shall show later, subgroups (Section 2.3) of these matrix groups provide a further wide range of examples. For instance, (a) by considering only those matrices with determinant 1 in GLm(F ) we obtain the m × m special linear group denoted by SLm(F ), or (b) by considering those matrices that have zeros at all entries below the main diagonal we obtain the group of m × m upper triangular matrices denoted by UTm(F ); see Section 3.3. Also many examples can be obtained by choosing different fields F .SoifF is finite, these matrix groups provide a variety of exam- ples of finite non-Abelian groups. For instance, SL2(F4) (which we usually write as SL2(4), the group of all 2 × 2 matrices A with det A = 1 and entries in the four element field F4—see page 254) is an important example of a simple group; as given by Definition 2.33. Many other simple groups can be defined using similar constructions; see Sections 12.2 and 12.3. Symmetries of Geometric Objects The symmetry properties of geometric objects provide a number of group examples. In Chapter 1 (page 3), we discussed the symmetries of an equilateral triangle, the 20 2 Elementary Group Properties group in question being called the dihedral group of the triangle denoted by D3.The elements of the group are the rotations and reflections that give the same geometric figure, and the operation is composition. A similar construction can be carried out for a regular polygon with n sides: the clockwise rotations about the centre are now by 2π/n, and ‘reflection’ or ‘turning over’ is as before. This group is denoted by Dn and again called dihedral. For example, D4 is the group of symmetries of the square, it has order 8. (Note that a few authors write D2n for Dn, see page 303.) Some other regular geometric objects have non-neutral (rotational) symmetry groups, for example, the tetrahedron (A4, see Problem 3.10), the octahedron (S4, see page 170) and the dodecahedron (A5, see Section 3.2 and Web Section 3.6). Also under this heading is the topic of sphere packing in various dimensions. Consider a large container filled with identical balls, some will touch adjacent balls and some will not. In dimension 2, where we have identical discs, a regular pattern forms and the set of disc centres gives a lattice (of equilateral triangles), and we can consider the symmetries of this lattice just as we have done for the triangle. In dimension 3, no such regular pattern forms where all adjacent balls touch. In this case, there are infinitely many ways to fit twelve balls around a central ball all touching it (there is always some room to spare), but thirteen never quite fit. In dimensions 8 and 24, ‘regular touching’ patterns do again form, the lattices given by the centres of the ‘spheres’ have some remarkable properties and give rise to some remarkable groups. For further details, see Conway and Sloane (1993). As a preliminary to this you should consider the following. The kissing number for these lattices is the maximum number of spheres that can fit around a central sphere S so that every sphere touches (kisses) S. In dimension 2, the kissing number is well- known to be six, and in dimension 3 it is, as noted above, twelve with some room to spare. But in dimension 8 it is 240, and in dimension 24 it is 196560 ! The first of these lattices has connections with the Mathieu group M24, and the second with the sporadic group called the ‘Monster’ or ‘Friendly Giant’, see Chapter 12,theATLAS (1985), and the reference quoted above. Permutations Permutations play a vital role in group theory, especially in the early development. If X is a set and SX denotes the collection of all permutations on X (that is, bijections of X to itself), then this collection forms a group under the operation of composition called the symmetric group on X.IfX is finite with n elements, we usually take X to be the set {1, 2,...,n} and write Sn for SX. See page 12 for the case n = 3. Note that Sn is non-Abelian if n>2, and has order n! (count all possible maps). As with many other groups, the symmetric groups have a number of important subgroups, that is, subsets that form groups; see Definition 2.11. For example, the alternating group An which is the group contained in Sn of all even permutations (a permutation is even if it can be expressed as an even number of interchanges of pairs of symbols; see Section 3.1) 2.2 Examples 21 Examples from Analysis Some classes of functions form groups. For example, let Z denote the set of all continuous, strictly-increasing functions f which map [0, 1] onto [0, 1], and satisfy f(0) = 0 and f(1) = 1. This set Z forms a group if the operation is taken to be composition of functions (the identity function f0, where f0(x) = x for all x, acts as the neutral element, and inverses exist as the functions f are continuous and strictly monotonic). We can construct further groups (subgroups) inside this one, for instance, we could consider only those functions in Z which are differentiable. These are examples of ‘topological groups’, see page 6. Free Groups and Presentations This construction provides another way to introduce groups, it will be discussed in more detail in Section 3.4 and Web Section 4.7. Consider an alphabet of letters A ={a,a,b,b,...}. The letter a is going to act as the inverse of a, et cetera, see page 57.Aword c1c2 ···ck consists of a finite string of letters ci from the alphabet A, for example, aabb a ,b,or ababa are words. The set of words with the operation of concatenation forms a semigroup; to obtain a group we proceed as follows. We define a reduced word as a word in which all pairs of consecutive letters aa,aa,bb,... do not occur or have been removed, for example, aabba reduces to a, whilst b and ababa are reduced. As a will act as the inverse of a, et cetera, each of these removals corresponds to the use of axiom (iv) in Definition 2.2. The operation of the group is as for the semigroup, that is concatenation—write one reduced word and then write the second reduced word immediately following the first, except that the resulting word must be reduced by removing all pairs aa,aa,bb,... if they are formed by the concatenation, or by previous removals. For instance, the product of ac b and b ca c is c. The empty word—that is the word with no symbols from A which is written as e where e/∈ A—acts as the neutral element of the group, and inverses are con- structed as in the example above—for instance, the inverse of aabcbc is cbcbaa. The group is denoted by a,b,c,... (in this notation it is assumed that the letters e,a,b,...are also present), and the letters a,b,c,...are the generators. It is called free because there are no constraints on possible words other than those ensuring the group properties hold; note that all free groups are necessarily infinite. A free group with just one generator a, say, is called an infinite cyclic group, it is isomorphic to Z and so we denote it either by a or by Z. Non-free groups have more condi- 22 2 Elementary Group Properties tions called relations, or sometimes defining relations. For example, the finite cyclic group Cn of order n can be treated as the infinite cyclic group Z a with the extra relation an = e. In this group, each time an occurs it is replaced by the neutral element e in the same way that terms of the form aa or aa are replaced by e. In Section 3.4, we shall see that this method for constructing groups has a number of advantages, but also some disadvantages. For instance, in a few cases it may be difficult, or sometimes impossible, to determine the group order. Elliptic Curves The collection of solutions of some equations can be formed into groups. For exam- ple, consider the set of rational solutions of the equation y2 = x3 + k where k ∈ Z\{0}. (2.2) Suppose P1 = (x1,y1) and P2 = (x2,y2) lie on the curve. The straight line P1P2 passing through the points P1 and P2 meets the curve in exactly one further point P3 = (x3,y3), say, and the pair (x3,y3) forms a new solution of (2.2), and if x1,...,y2 ∈ Q then also x3,y3 ∈ Q.IfP1 = P2, then the line is the tangent to the curve at P1; and the whole procedure is called the chordÐtangent process. Points on the curve with rational coordinates form the elements of a group, and the operation is defined using the chord–tangent process. It is closed because, given rational points P1 and P2 on C, a rational point P3 on C always exists, and we set P1 + P2 =−P3. The neutral element is the ‘point at infinity’ on the curve in the y-direction. (To set this procedure up properly we use homogeneous coordinates (x : y : z), where ∗ (tx : ty : tz)= (x : y : z) for all t ∈ Q . The usual notation for a point (x, y) is identified with (x : y : 1), and the point (x : y : 0) lies on the ‘line at infinity’. Equation (2.2) becomes y2z = x3 +kz3. The point (0 : 1 : 0), the neutral element of the group, clearly lies on the curve, and is the ‘point at infinity’ in the y-axis direction. We set P1 + P2 to equal ‘minus’ P3 to obtain a valid inverse operation, so using the standard two variable (affine) coordinates, the inverse of the point P1 = (x1,y1) is −P1 = (x1, −y1).) In this ‘projective geometry’ all vertical lines ‘meet’ at the point at infinity (0 : 1 : 0), and some results from algebraic geometry are needed to prove associativity. These groups can be finite or infinite, and they are Abelian because the line through the points P1 and P2 is clearly the same as the line through P2 and P1. See for example Rose (1999), Chapters 15 and 16, for further details. 2.2 Examples 23 Examples from Topology The basic structure of a topological space is best described using groups. For exam- ple, the fundamental group of a space, which was first defined by H. Poincaré over a century ago, is constructed as follows: Fix a point P in a path-wise connected topological space T , and consider the set of all continuous closed and directed loops from P to P . Call two loops ‘equivalent’ if one can be continuously deformed into the other (topologists call them ‘homo- topic’), the group operation is composition. The neutral element is the set of loops that can be continuously contracted to the point P , and the inverse of the loop L is L with its direction reversed. For instance, the fundamental group of the real plane R2 with the origin removed is the infinite cyclic group (all loops through P that do not enclose the origin can be contracted to P , but, for example, a loop that passes around the origin clockwise four times, say, cannot be contracted and ‘equals’ four times a loop which passes around the origin clockwise only once). Although the definition appears to depend on the point P ∈ T , it can be shown that fundamental groups for different points P are isomorphic; that is, there exists a fundamental group for the space T . Further details can be found in a standard book on general topology, for example, Kelley (1955) or Willard (1970). Algebraic topology is another area where groups—homology and cohomology groups-provide insights into the structure of topological spaces, see for example Benson (1991). Examples from the Physical Sciences Particle physicists make extensive use of group theory. Many of the essential prop- erties of the basic constituents of matter are best described using the language and properties of groups. At least one elementary particle was discovered using the abstract theory. A collection of particles was associated with a particular class of groups, and it was realised that there was one more group (that is, 28) than known particles in this collection (at the time, 27); this led the experimental workers to look for the ‘missing’ particle (called Ω−), and it was duly found a few years later! An excellent ‘down-to-earth’ account of this topic is given in Close (2006), and Williams (1994) provides a good technical introduction. Some chemists use groups to describe the structure of molecules. A notable ex- ample was given in 1985 when a crystalline form of carbon, called Carbon60, was discovered by the chemists Kroto, Curl and Smalley;2 its structure is closely related to that of a dodecahedron, and so also to the alternating group A5; the subsection on symmetries above, Section 3.2, and Web Section 3.6 all give some details. 2They were awarded the Nobel prize in chemistry for this work. 24 2 Elementary Group Properties 2.3 Subgroups, Cosets and Lagrange’s Theorem Most groups contain a number of smaller groups using the same operation, we shall consider these now. Definition 2.11 A subgroup H of a group G is a non-empty subset of G which forms a group under the operation of G. We write H ≤ G when H is a subgroup of G. For example, if G is the group Q, then Z is a subgroup, that is, Z ≤ Q. But note that Q+ is not a subgroup of Q even though the underlying set in the first group is contained in the second; reader, why? Definition 2.12 (i) A subgroup J of a group G is called proper if J = G,thisis denoted by J Notes (a) The neutral subgroup e is sometimes called the identity, trivial, or unit, subgroup; see page 4. (b) Clearly G ≤ G; so all groups with more than one element have at least two subgroups; some have no more, see Theorem 2.34. (c) Maximal subgroups are not necessarily ‘large’. For an extreme example, con- sider the alternating group A13 which has order 3113510400, remarkably it pos- sesses a maximal subgroup of order 78. Also arbitrarily large groups with maximal subgroups of order 2 exist—Problems 3.20 and Corollary 6.12. (d) There are connections between maximal subgroups and generators, see Prob- lem 2.13 and Section 10.2, and reasoning with maximal subgroups is used in several proofs, for example, in that for the Frattini Argument (Lemma 6.14). The next result gives conditions for a group subset to be a subgroup. Theorem 2.13 If H is a subset of G, then H ≤ G if, and only if, (a) H is non-empty, and (b) whenever a,b ∈ H , we also have a−1b ∈ H . Proof Clearly (a) and (b) are valid if H ≤ G (Definitions 2.2 and 2.11). Conversely, suppose (a) and (b) hold for a subset H of G. By (a), there is at least one element a ∈ H , and so, by (b), e = a−1a ∈ H . Applying (b) again, we have, as a,e ∈ H , a−1 = a−1e ∈ H , and so H is closed un- der inverses. Thirdly, if a,b ∈ H , then a−1 ∈ H , and so together these give 2.3 Subgroups, Cosets and Lagrange’s Theorem 25 ab = (a−1)−1b ∈ H by Theorem 2.7(ii), that is, H is closed under the opera- tion of G. Finally, we note that associativity holds in H because it holds in G; the result follows. There is also a ‘right-hand version’ of this result where in (b) ‘a−1b ∈ H ’issubsti- tuted by ‘ab−1 ∈ H ’. In practice, it is often better to replace (b) by: (b1) if a,b ∈ H then ab ∈ H , and (b2) if a ∈ H then a−1 ∈ H . For example, suppose G = GL2(Q) and H is the subset of these matrices with de- terminant 1. The identity matrix I2 belongs to H , and so H is not empty. Also if A,B ∈ H , then det A = det B = 1 and so, as det(AB) = det(A) det(B) = 1, we de- duce AB ∈ H . Finally, if C ∈ H , then 1 = det C = det C−1, and so C−1 ∈ H ;this gives H ≤ G. We use the notation SL2(Q) for H , see Section 3.3. Some further examples are given in Problem 2.10. We consider now some set-theoretic operations on subgroups. Corollary 2.14 (i) If H ≤ J and J ≤ G, then H ≤ G, that is the subgroup relation is transitive. (ii) If H,J ≤ G and H ⊆ J , then H ≤ J . Proof Both of these results follow from Theorem 2.13, see Problem 2.5. Intersections of subgroups are always subgroups (but unions are usually not sub- groups because closure fails). See the note concerning subgroup lattices on page 32. Theorem 2.15 Suppose I is a non-empty index set. If Hi ≤ G, for each i ∈ I , and = ≤ J i∈I Hi , then J G. Proof As e ∈ Hi for all i ∈ I ,wehavee ∈ J ,soJ is not empty. Secondly, if a,b ∈ J , then a,b ∈ Hi for all i ∈ I , but each Hi ≤ G so, by Theorem 2.13, −1 −1 a b ∈ Hi , for all i ∈ I , which shows that a b ∈ J . Now use Theorem 2.13 again. In Section 2.1 (page 13), we introduced the notion of a generating set for a group, this can be formally defined by Definition 2.16 A subset X of the underlying set of a group G is said to generate G if the intersection of all subgroups of G that contain X coincides with G,or to put this another way, the only subgroup of G that contains X is G itself. This intersection is denoted by X. Theorem 2.17 Suppose X is a non-empty subset of the underlying set of the group G. The set X generates G if and only if the set of all products of powers (positive and negative) of elements of X equals G. 26 2 Elementary Group Properties Proof By Theorem 2.15 and Definition 2.16, X≤G. Suppose X=G. Let Z denote the set of all powers of products of elements of X; Z ⊆ G). The set Z is non-empty as X is non-empty, and so Z ≤ G by Theorem 2.13 and the definition of Z.AlsoX ⊆ Z, and so Z is one of the subgroups used in the formation of the intersection X; hence X≤Z ≤ G. Therefore, as X generates G (by supposition), we have Z = G. For the converse suppose Z = G.NowforgivenH ,ifX ⊆ H and H ≤ G, then Z ≤ H , again by Theorem 2.13 and the definition of Z. This holds for all such H , and so it holds for X by Theorem 2.15; that is, Z ≤X. But by our supposition Z = G, and so X=G, and the result is proved. We set X=e,ifX is empty. Now we consider group elements in more detail. For g ∈ G we write g for the setofpowersofg ∈ G, that is g={gt : t ∈ Z}; see Section 4.3.Wehave Theorem 2.18 If g ∈ G then g≤G. Proof The set g is clearly not empty, and if m, n ∈ Z, then gm,gn ∈g, and (gm)−1gn = gn−m ∈g. Result follows by Theorems 2.9 and 2.13. We say that g is a generator of the subgroup g of G (page 13). This result ensures that almost all groups have at least some non-neutral proper subgroups, see Theorem 2.34 for the exceptions. Examples (a) Let G = Z and g = 7, then 7 is the proper subgroup of Z consisting of the set of integers divisible by 7. (b) Secondly, let G = (Z/7Z)∗ and g = 3. In this case, the subgroup 3 is G itself because the powers of 3 modulo 7 generate the whole group; the reader should check this and also consider the case g = 2. Theorem 2.18 and these examples suggest the following Definition 2.19 Let g ∈ G. (i) The subgroup g given by Theorem 2.18 is called cyclic. (ii) The order of g, denoted by o(g), is defined by o(g) = o(g); that is, o(g) equals the order of the cyclic subgroup generated by g in G. (iii) An element of order 2 is called an involution. (iv) The exponent, if it exists, of a group G is the least common multiple of the orders of all of the elements of G; that is, the least positive integer m with the property: gm = e for all g ∈ G. Notes (a) All parts of this definition are relative to a fixed group G. (b) Orders can be finite or infinite, and if the orders of two elements are finite it does not follow that the order of their product is finite (Problem 2.7). 2.3 Subgroups, Cosets and Lagrange’s Theorem 27 (c) If G is finite, it has an exponent which is not greater than o(G). The group Q is an example of an infinite group with no exponent. In 1902, W. Burnside (1852– 1927) conjectured that a group G with finite generating set and finite exponent must be finite, and this is true if G is Abelian. But it can be false if G is not Abelian as was shown by Adian and Novikov in 1968 for a group with an exponent larger than 665; see Vaughan-Lee (1993). (d) Elements of order 2 are called involutions to signal the fact that they play a unique role in the theory, particularly to CFSG. (It is the subgroups called cen- tralisers of these involutions, see Section 5.2, that play this vital role.) Also, apart from the neutral element e they are the only group elements which equal their own inverses. Further properties are given in Problems 2.28, 3.1(iv) and 3.20, and in the note about Coxeter Groups on page 64. We illustrate these concepts with the following result. Corollary 2.20 If a group G has exponent 2, then it is Abelian. Proof Suppose a,b ∈ G, then ab ∈ G and e = (ab)2 = abab. Multiplying on the left by a and on the right by b, we obtain ab = aeb = a(abab)b = a2bab2 = ba as both a and b have order 2. This holds for all a,b ∈ G. Given a prime p, an Abelian group all of whose non-neutral elements have order p is called an Elementary Abelian p-group. We shall see later (Problem 4.18) that these groups can be treated as vector spaces defined over a p-element field. The corollary above shows that all groups of exponent 2 are of this type, this is not true for primes p>2; an example is given in Problem 6.5. Cosets and Lagrange’s Theorem For our next results, we require some new notation. If X and Y are non-empty subsets of a group G, then we write XY for the subset XY ={xy : x ∈ X and y ∈ Y }⊆G. If X is the singleton set {x}, then we write xY for {x}Y (and Yx for Y {x}). Note that if X, Y, Z ⊆ G then, by associativity, (i) X(YZ) = (XY )Z, and (ii) XY = YX,ifG is Abelian. Definition 2.21 For H ≤ G and g ∈ G,thesetgH ={gh : h ∈ H } is called a left coset of H in G, and the set Hg is called a right coset of H in G. 28 2 Elementary Group Properties Cosets play an important role in the theory, here they lead to our first major result—Lagrange’s Theorem, and in Chapter 4 they form part of the important ideas associated with factor groups. One of the origins of this work is Gauss’s develop- ment of modular arithmetic undertaken two centuries ago (page 18): if G = Z and H = nZ, then the cosets are kH ={k + nz : z ∈ Z} for k = 0,...,n− 1; the coset kH equals the set of integers congruent to k modulo n. When referring to the set T of cosets of H in G, we often write T ={gH : g ∈ G}. Here we are using the convention that in an un-ordered set duplication is ignored, for instance, the set {...,a,a,...,a,...} is the same as {...,a,...}.Ifwe did not use this convention in the coset case, we would need to specify a unique g in each coset gH, and this would cause problems. We begin with some basic lemmas. The first will be used often in the following pages, it characterises the coset representatives. Lemma 2.22 If H ≤ G and a,b ∈ G, then aH = bH if and only if a ∈ bH if and only if b−1a ∈ H. There is an exactly similar result for right cosets; the reader should write it out and redo the following proof in this second case. Proof Suppose firstly aH = bH.AsH is a subgroup, e ∈ H and so a = ae ∈ aH = bH. Secondly, suppose a ∈ bH, then there exists h ∈ H satisfying a = bh, and so b−1a = h ∈ H . Lastly, suppose b−1a ∈ H . As above this gives a = bh for some h ∈ H , and hence ah1 = bhh1 ∈ bH for all h1 ∈ H ; that is, aH ⊆ bH. For the converse inclusion, as H ≤ G, we have by Theo- rem 2.13, a−1b = (b−1a)−1 ∈ H , and so we can repeat the previous argument with a and b interchanged, the equation aH = bH follows. To derive Lagrange’s Theorem, we require the following three lemmas, the first shows that cosets are either disjoint or identical. Lemma 2.23 If H ≤ G, then the underlying set of G can be expressed as a disjoint union of the collection of all left cosets of H in G. There is an exactly similar result for right cosets. Proof Clearly, each element g ∈ G belongs to a left coset because g ∈ gH. Suppose further g ∈ aH and g ∈ bH, then by Lemma 2.22, aH = gH = bH, and the lemma follows. The right coset version follows similarly. 2.3 Subgroups, Cosets and Lagrange’s Theorem 29 Lemma 2.24 If H ≤ G and g ∈ G, then o(H) = o(gH) = o(Hg). Proof We give the proof for left cosets, the right coset result is proved sim- ilarly. To establish this lemma, we construct a bijection between the sets in- volved. Let φ be the map from H to gH defined by hφ = gh; see note on ‘left or right’ on page 68.Ifhφ = hφ then gh = gh, and by cancellation (Theorem 2.6) this gives h = h, and so φ is injective, hence it is bijective because it is clearly surjective. Lemma 2.25 If H ≤ G, then the number (cardinality) of left cosets equals the num- ber of right cosets. Proof As in the previous proof, we construct a bijection between the sets. Let θ be the map from the set of left cosets to the set of right cosets given by (gH )θ = Hg−1. = −1 = −1 This is well-defined; for if gH g1H , then Hg Hg1 (use left and right versions of Lemma 2.22 and closure under inverses). It is also clearly surjective because each element of G is the inverse of some element in G (Theorem 2.8). To prove injectivity, suppose = −1 = −1 (gH )θ (g1H)θ, that is Hg Hg1 . −1 ∈ −1 Using the right-hand version of Lemma 2.22, this gives g1 Hg , and −1 = −1 ∈ = −1 ∈ hence g1 hg for some h H . Therefore, g1 gh gH (as H is a subgroup), and so g1H = gH by Lemma 2.22 again. Having established these lemmas, we can now derive Lagrange’s Theorem. Much of the early work in algebra was concerned with properties of polynomials defined over the rational numbers. J.-L. Lagrange (1736–1813), an Italian mathematician working in France, studied these polynomials as well as in many other topics in mathematics. He postulated a result similar to that given in the last part of Prob- lem 5.1 which relies on what we now call ‘Lagrange’s Theorem’; and it is for this reason that the following result is so named. In fact, Galois gave the first proof of the theorem for permutation groups in 1832, and it was probably C. Jordan (1838–1932) who gave the first proof for general groups some thirty years later. We begin by making the following Definition 2.26 Let H ≤ G. The number (cardinality) of left cosets of H in G is called the index of H in G, it is denoted by [G : H]. By Lemma 2.25, this equals the number of right cosets of H in G. 30 2 Elementary Group Properties Theorem 2.27 (Lagrange’s Theorem) If H ≤ G then o(G) = o(H)[G : H ]. Proof By Lemma 2.23, the underlying set of G is a disjoint union of [G : H ] cosets, and by Lemma 2.24, each of these cosets has the same cardinality (number of elements), that is o(H), the theorem follows. This result is particularly useful in the finite case where it shows that H can only be a subgroup of G if o(H) | o(G); that is, the prime factorisation of the order of a group G is an important invariant of G. For instance, a group of order 30 cannot have subgroups of order 4, 7, 8, 9, 11,...,29. Also, it cannot have elements of order 4, 7,..., see Definition 2.19. In the infinite case, the theorem shows that either the order of the subgroup, or the index (or both), must be infinite. Note that there exist infinite groups all of whose proper subgroups are finite, see Problem 6.7. 2.4 Normal Subgroups The last topic in this chapter concerns a special type of subgroup in which left and right cosets are equal, they play a vital role in the theory. These subgroups were first defined by Galois in the 1820s when he was working on the solution of polynomial equations by radicals; see the Introduction to Chapter 11. Definition 2.28 (i) Let K ≤ G. The subgroup K is called normal in G if, and only if, gK = Kg for all g ∈ G. This is denoted by K G. (ii) If g,h ∈ G, h−1gh is called the conjugate of g by h in the group G. (iii) For a fixed element g ∈ G,theset{h−1gh : h ∈ G}, that is, the set of conju- gates of g in G, is called the conjugacy class of g in G; see also Definition 5.17. Notes (a) The subgroups e and G are normal in G for all groups G. (b) If G is Abelian, all subgroups are normal, all conjugates of g equal g, and so the conjugacy class of g in G is {g}. (c) We reserve the symbol ‘K’, possibly with primes or subscripts, to denote a normal subgroup, but other symbols will occasionally be used where necessary. In Chapter 4, we discuss the connection between normal subgroups and kernels of homomorphisms—K for kernels and so also for normal subgroups. (d) Conditions stronger than normality are useful at times. The first is character- istic, for a definition and basic properties see Problem 4.22; the main point is that characteristic is a transitive property whereas normality is not. Some authors use an even stronger property called fully invariant which is defined similarly to character- istic except that in the definition on page 89 the word ‘automorphism’ is replaced by ‘endomorphism’, see Definition 4.2. 2.4 Normal Subgroups 31 The following theorem gives two conditions for normality, see note below the statement of Lemma 4.6. Theorem 2.29 (i) If K ≤ G, then the following conditions are equivalent: (ia) K G; (ib) for all g ∈ G, g−1Kg ⊆ K; (ic) for all g ∈ G and all k ∈ K, g−1kg ∈ K. (ii) Suppose K G. If k ∈ K, then all conjugates of k in G belong to K, and K is the union of a collection of the conjugacy classes of G. Proof Note first that both parts of (ii) follow immediately from (i). Suppose (ia) holds, so if g ∈ G, gK = Kg by definition. Hence, for all k ∈ K, we can find k ∈ K to satisfy − gk = kg, that is g 1kg = k ∈ K, which gives (ib). Secondly, note that (ic) follows immediately from (ib) (as g−1kg ∈ g−1Kg). Finally, suppose (ic) holds. So if g ∈ G and k ∈ K, we can find k ∈ K to satisfy − g 1kg = k , which gives kg = gk and so Kg ⊆ gK, as this argument holds for all k ∈ K. For the converse, we have gkg−1 = (g−1)−1kg−1 ∈ K, and so we can find k ∈ K to satisfy gkg−1 = k or gk = kg. This gives the reverse inclusion and (ia) follows. Notes To prove that K G it is necessary to prove both K ≤ G and K is normal in G. Secondly, normality is not transitive (cf. Corollary 2.14); that is, if K G and H K, it does not follow that H G; see Problem 2.19(iii) for an example. On the other hand, if K G and K ≤ H ≤ G, then K H ; see Problem 2.14. A stronger property called characteristic which is transitive was mentioned in (d) opposite. Our first application of the normal subgroup concept answers the question: When is the product HJ of two subgroups H and J itself a subgroup? Note that in general HJ is not a subgroup because it is not closed under the group operation. We use the notation H ∨ J (or sometimes H,J)—the join of H and J —for the group generated by the elements of both H and J (Definition 2.16). Clearly HJ ⊆ H ∨ J , we have Theorem 2.30 Suppose H,J ≤ G. (i) If either H or J is a normal subgroup of G, then HJ ≤ G and H ∨ J = HJ = JH. (ii) If both H G and J G, then HJ G. 32 2 Elementary Group Properties Proof (i) Suppose hi ∈ H,ji ∈ J , i = 1, 2, and H G (the proof is similar −1 −1 = ∗ ∗ ∈ −1 ∈ ∈ if J G). Then j1 (h1 h2)j1 h for some h H (as h1 h2 H,j1 J and H G). Hence −1 = −1 −1 −1 = ∗ −1 ∈ (h1j1) (h2j2) j1 h1 h2j1j1 j2 h j1 j2 HJ, and, as HJ is clearly not empty, HJ ≤ G follows by Theorem 2.13.Asim- ilar argument shows that a product of terms each of the form hj ,forh ∈ H and j ∈ J , is itself of this form; that is, HJ = H ∨ J . The last equation in (i) follows because H ∨ J = J ∨ H , or we can show directly as above that HJ ⊆ JH and JH ⊆ HJ. (ii) By (i), we only need to check normality. If g ∈ G, h ∈ H and j ∈ J , we have g−1hj g = g−1hgg−1jg ∈ HJ by hypothesis, the result follows. Subgroup Lattices Using Corollary 2.14 and Theorems 2.15 and 2.30, the collection of subgroups of a group forms a (complete) lattice L, that is, a non-empty partially ordered set (Def- inition A.5 in Appendix A) in which every subset has a greatest lower bound and a least upper bound in L. Note that both the intersection and the join of two sub- groups of a group G are themselves subgroups of G. Some examples are given in Chapter 8. The structure of this lattice can have an important bearing on the group in question. The first major result (Ore 1938) states that the lattice of a finite group G is distributive (that is, H ∨ (J ∩ K) = (H ∨ J)∩ (H ∨ K), et cetera.) if and only if G is cyclic. It should be noted that non-isomorphic groups can have identical sub- group lattices; see Problem 6.4. For a detailed account of this aspect of the theory, the reader should consult Schmidt (1994). The centre of a group is an important example of a normal subgroup, it is given by Lemma 2.31 In a group G, the set J ={a ∈ G : ag = ga for all g ∈ G} forms a normal Abelian subgroup of G. Proof Suppose a ∈ J . For all g ∈ G,wehaveeg = ge, ag = ga im- plies a−1g−1 = g−1a−1 by Theorem 2.7, and if ag = ga and bg = gb then abg = agb = gab; and so J ≤ G by Theorem 2.13.AlsoJ is Abelian by definition. Lastly, note that ag = ga implies g−1ag = a, and so J G by Theorem 2.29. Definition 2.32 The subgroup J of G giveninLemma2.31 is called the centre of G, it is denoted by Z(G). 2.4 Normal Subgroups 33 Notes The notation Z(G) is used because German authors call this subgroup the Zentrum. The centre of a group G gives some important information about G. Clearly, Z(G) = G if and only if G is Abelian. On the other hand, some groups are centreless, that is, Z(G) =e; examples are D3 and S4, see Problem 2.26. A centreless group can in some ways be treated as the opposite of an Abelian group. We end this chapter by introducing simple groups. We shall show later they can be treated as the basic ‘building blocks’ for the construction of all finite and some infinite groups; see Chapter 9. Definition 2.33 A group G is called simple if it contains no proper non-neutral normal subgroup. The term ‘simple’ is perhaps not well-chosen because some simple groups are very complicated! But as noted above they can be used as the basic constituents of all groups; of course, they are all centreless. A full list of finite simple groups is now known; see the ATLAS (1985) and Chapter 12. Many simple groups are given by Lagrange’s Theorem for we have Theorem 2.34 If o(G) is a prime number, then the group G is simple and cyclic. For the converse see Theorem 9.6. Proof By Lagrange’s Theorem (Theorem 2.27), the order of a subgroup of G divides o(G). But in this case, the only positive divisors of the integer o(G) are 1 and p, hence G has no proper non-neutral subgroups at all, and so clearly no proper non-neutral normal subgroups. Also by Theorem 2.18 and Definition 2.19, every element has order 1 or p. There is only one element, e, of order 1 (Lemma 2.4). Hence G has p − 1 elements of order p;leta be one of them. As a has p distinct powers (including p0 = e), it follows that all elements of G equal powers of a, and so G is cyclic. In fact, ‘most’ simple groups (counted by the size of their orders) are of this type, that is Abelian (and cyclic). For example, there are 173 (isomorphism classes of) simple groups with order less than 1000 but only five are non-Abelian. The construction of non-Abelian simple groups is a much more difficult task, in the next chapter we introduce the first groups of this type—alternating groups, and more will be discussed in Chapter 12. These include a number of infinite classes of matrix groups, especially the linear groups Ln(q) and the unitary groups Un(q), and also 26 (!) so called sporadic groups. These groups range in size from 7920 84 (Mathieu group M11) to about 10 (Friendly giant M) and they have a wide variety of constructions. The existence of these non-Abelian simple groups is surely one of the most interesting and challenging aspects of the theory. 34 2 Elementary Group Properties 2.5 Problems A number of the problems given below have important applications in the sequel. For an explanation of the symbols and , see page 9. Problem 2.1 (i) Write out a proof of Theorem 2.8. (ii) Using induction on n, prove the generalised associativity law for groups: If g1,...,gn ∈ G, then all expressions formed by inserting or deleting brackets (in corresponding pairs) in the term g1 ··· gn are equal. Problem 2.2 Show that the following sets with operations form groups, and indi- cate which are Abelian. (i) Z/7Z with addition modulo 7. (ii) (Z/7Z)∗ with multiplication modulo 7. (iii) The set Q with the operation ∗ where, for a,b ∈ Q,wehavea ∗ b = a + b + 3. (iv) GL2(Q) with matrix multiplication, see also Problem 2.10 below. (v) The set of powers of products Q (that is, the group generated by) of the com- A = 01 B = 0 i plex matrices −10 and i 0 with the operation of matrix multi- plication. What is the order of this group? (vi) Let R = R ∪{∞} where the symbol ∞ satisfies the usual naive rules: 1/0 =∞,1/∞=0, ∞/∞=1 and 1 −∞=∞=∞−1. Define six func- tions mapping R onto itself by: 1 f (x) = x, f (x) = ,f(x) = 1 − x, 1 2 x 3 1 x x − 1 f (x) = ,f(x) = ,f(x) = . 4 1 − x 5 x − 1 6 x Show that this set forms a finite group under the operation of composition. (vii) Let R denote the real plane R2,letd denote the standard distance function (metric) on R, and let denote the set of bijective maps of R to itself which preserve distance—if x,y ∈ R and θ ∈ , then d(x,y) = d(θ(x), θ(y)). A function of this type is called an isometry; rotation by π/3 about the origin is an example. Show that with the operation of composition forms a group. The reader needs to be convinced that all the sets with operations described in Section 2.2 are, in fact, groups. Problem 2.3 Why are the following sets with operations not groups? (i) The integers Z with subtraction. (ii) The set of odd integers with addition. ar ∪ c 0 ∈ R = = (iii) The set 0 b sd where a,b,...,s and ab 1 cd, with matrix multiplication. (iv) The rational numbers Q with multiplication. 2.5 Problems 35 Problem 2.4 (i) Let S be a semigroup with cancellation, so it has closure under its operation which is associative, and for all a,b ∈ S, we can find x,y ∈ S to solve the equations ax = b and ya = b. Show that S forms a group. (ii) If T is a semigroup, and for all a ∈ T there is a unique a∗ ∈ T satisfying aa∗a = a, prove that T is a group. Problem 2.5 If H,J ≤ G and in (iii) p is a prime, show that (i) If H is a subset of J , then H ≤ J . (ii) H ∩ J = H if, and only if, H ≤ J . (iii) If o(H) = o(J) = p, then either H = J or H ∩ J =e. Problem 2.6 Prove that if G is a group and S ≤ G, then SS = S. Conversely, if T is a non-empty finite subset of G and TT = T , prove that T ≤ G.IsthistrueifT is infinite? Problem 2.7 (Order Function) Let g,h ∈ G. Prove the following properties of the order function. (i) o(gh) = o(hg). (ii) If o(g) = n and m ∈ Z, then o(gm) = n/(m, n); see page 284. (iii) If o(g) = m and (m, n) = 1, there exists h ∈ G satisfying hn = g. (iv) If o(g) = m, o(h) = n, and g and h commute, then o(gh) = LCM(m, n);see part (vii). (v) If G is finite and g ∈ G, then go(G) = e, and o(g) | ex(G) where ex denotes the exponent of G, see Definition 2.19. (vi) Suppose g ∈ G and o(g) = mn where (m, n) = 1. Show how to find unique a,b ∈ G to satisfy ab = g = ba, o(a) = m and o(b) = n. (vii) In (iv), if we drop the commutativity condition show that o(gh) can be infinite. (Hint. Try G = GL2(Q).) Problem 2.8 (i) Suppose G is a finite group and o(G) is even. Is the number of elements of order 2 in G odd—does a group of even order always contain an invo- lution? See also Cauchy’s Theorem (Theorem 6.2). (ii) Using the group (Z/pZ)∗ where p is prime, see the definition on page 18, give a proof of Fermat’s Theorem: − ap 1 ≡ 1 (mod p) if (a, p) = 1. (iii) Using the same group as in (ii), show that (p − 1)!≡−1 (mod p),are- sult sometimes (wrongly) known as Wilson’s Theorem. You are given: If p>2, then (Z/pZ)∗ contains exactly two elements of order at most 2 (Theorem B.13 in Appendix B). 36 2 Elementary Group Properties Problem 2.9 (Multiplication Tables) Given a group G of order n with elements g1,...,gn where g1 = e, we can form a square array or table, with n rows and n columns, whose (i, j)th entry is the product gigj . Show that each row and each column of this table is a permutation of the elements g1,...,gn. What can you say about the first row and first column? Is the converse true? That is, if we have a square array of elements such that each row and each column is a permutation of some fixed set, and the first row and column have the property mentioned above, does the corresponding array always form the multiplication table of a group? Problem 2.10 Show that the following subsets are subgroups of the corresponding groups, and determine whether they are normal. (i) The set {1, −1} in R∗. (ii) The set of permutations on Y ={1,...,6} which leave 3 fixed in S6,thesetof all permutations on Y ; see Section 3.2. (iii) The subsets of GL2(Q) of matrices (a) which have determinant 1, and (b) which are upper triangular (that is, the bottom left-hand entry is zero); see page 19 and Section 3.3. (iv) The set of complex numbers with absolute value 1 in C∗. (v) The set of differentiable functions in the group Z described in the subsection on groups in analysis on page 21. Problem 2.11 (i) Show that a finite subgroup of the multiplicative group of the complex numbers C∗ is cyclic. (Hint. Consider roots of unity.) (ii) Find as many subgroups as you can of the additive group of the rational numbers Q;seeWeb Section 7.5. Problem 2.12 List the left and right cosets of the subgroups given in Problem 2.10; note that the last part is not easy! Problem 2.13 (i) Can a subset of a group G be the left coset of two distinct sub- groups of G? (ii) If G is finite and has a unique maximal subgroup H , show that it is cyclic. (Hint. Consider an element in G\H .) Problem 2.14 (Normality Properties) Prove the following statements—all widely used in the sequel. (i) If K G and K ≤ H ≤ G, then K H . (ii) A subgroup of the centre of a groupG is normal in G. = n (iii) If Ki G for i 1, 2,...,n, then i=1 Ki G. (iv) If H,J,K ≤ G and K J , then K ∩ H J ∩ H . 2.5 Problems 37 Problem 2.15 (i) Show that if [G : H] is finite and H ≤ J ≤ G, then [G : H]=[G : J ][J : H ]. (ii) Prove that if H,J ≤ G with [G : H]=m and [G : J ]=n, then [G : H ∩ J ]≥ LCM(m, n), and equality occurs if, and only if, m and n are coprime. Problem 2.16 (Derived Subgroup) If g,h ∈ G we set [g,h]=g−1h−1gh,itis called the commutator of g and h. Also the subgroup of G generated by the set of all products of powers of the commutators of G is called the derived (or commutator) subgroup of G, and it is denoted by G. In some cases, the set of commutators of a group does, in fact, form a subgroup of the group, but not always; for an example, seeRotman(1994), page 34. More generally, if H,J ≤ G,welet[H,J] denote the subgroup generated by all commutators of the form [h, j] where h ∈ H and j ∈ J ; so, for example, [G, G]=G. See also Problem 4.6(ii), and Section 11.1, especially page 234. (i) Show that G G. (ii) Find G when G is (a) Z,(b)D3, and (c) Q, see Problem 2.2(v). (iii) Prove that if J ≤ G and J ⊇ G, then J G—an important fact with many applications. (iv) Show that if K G and K ∩ G =e, then K ≤ Z(G), and so in particular K is Abelian. (v) Finally, prove that if K G and J =[K,G], then J ≤ K and J G. Problem 2.17 (Commutator Identities) Prove the following identities where [a,b,c]=[[a,b],c] for a,b and c in the same group G. Identity (iv) is called the HallÐWitt Identity. (i) [b,a]=[a,b]−1, (ii) If a,b ∈ G, and both a and b commute with [a,b], show that [ar ,bs]=[a,b]rs for r, s ∈ Z, − (ab)t = at bt [b,a]t(t 1)/2 if t ≥ 0. (Use induction on t, (i), and the given relationship between G and Z(G).) (iii) [ab,c]=(b−1[a,c]b)[b,c] and [a,bc]=[a,c](c−1[a,b]c), (iv) b−1[a,b−1,c]bc−1[b,c−1,a]ca−1[c,a−1,b]a = e. (v) If a1,...,am,b1,...,bn ∈ G and H =a1,...,bn, then we can express [a1 ...am,b1 ...bn] as a product of conjugates of [ai,bj ] by some cij ∈ H . (vi) If H,J ≤ G where G =H,J, then [H,J] G. Problem 2.18 Suppose A,B,C ≤ G and A ≤ B. Show that (i) B ∩ (AC) = A(B ∩ C), (ii) if G = AC then B = A(B ∩ C), (iii) if AC = BC and A ∩ C = B ∩ C, then A = B. 38 2 Elementary Group Properties (Note that AC and/or BC may not be subgroups of G, also (i) and (ii) are sometimes known as Dedekind’s Modular Laws.) (iv) Now suppose A,B,C,D ≤ G where also AB, CD ≤ G. Show that if A ≤ D and C ≤ B then AB ∩ CD = AC(B ∩ D). Problem 2.19 Let H,J ≤ G. Prove the following results. (i) If [G : H ]=2, then (a) H G, and (b) a2 ∈ H for all a ∈ G—facts we use many times. (ii) If G is finite and o(H ) > o(G)/2, then H = G—no finite group can have a proper subgroup of order larger than half the group order. Further, if G is also simple and J ≤ G, then o(J) ≤ o(G)/3. For large simple groups, the denominator 3 can be replaced by a much bigger integer; see example below Theorem 5.15, page 101. (iii) Show that normality is not transitive (that is if H J and J G, it does not follow that H G); one example occurs in D4 using (i). (iv) If H and J are proper subgroups of G, prove that there exists g ∈ G which does not belong to either H or J . (v) Show that HJ ≤ G,ifHJ = JH; cf. Theorem 2.30(i). Problem 2.20 Using Corollary 2.20, Lagrange’s Theorem (Theorem 2.27) and Problem 2.5, show that up to isomorphism there are only two groups of order 4, and only two groups of order 6—that is, there are exactly two isomorphism classes of groups of order 4, and also exactly two of order 6. (Hint. For order 6, show that the group always contains an element of order 3.) See Problem 4.2(i), different methods to prove these facts are given in Chapters 5 and 6. Problem 2.21 Let G be a group. If Hi ≤ G and [G : Hi] is finite for i = 1,...,n, show that n n G : Hi ≤ [G : Hi]. i=1 i=1 (Hint. Derive the result for n = 2first.) Problem 2.22 (Poincaré’s Theorem) Prove that the intersection of a finite number of subgroups of G, each with finite index, is itself a subgroup of G with finite index. Problem 2.23 If H ≤ G and g ∈ G, then g−1Hg is called a conjugate subgroup of H (Definition 2.28). Prove the following statements: (i) g−1Hg ≤ G, (ii) o(g−1Hg)= o(H), (iii) g−1Hg ={j ∈ G : gjg−1 ∈ H}. 2.5 Problems 39 Problem 2.24 (Core of a Subgroup) If H ≤ G,thecore of H in G, core(H ),is defined by − core(H ) = g 1Hg, g∈G see Section 5.2. Show that (i) core(H ) G, (ii) core(H ) is the join of all normal subgroups of G which are contained in H , (iii) core(H ) is the unique largest normal subgroup of G contained in H . Problem 2.25 (Normal Closure of a Subgroup) If H ≤ G, then the normal clo- sure H ∗ of H is defined as the intersection of all normal subgroups of G which contain H . Show that (i) H ∗ G, (ii) H ∗ =g−1Hg : g ∈ G, (iii) H ∗ is the smallest normal subgroup of G containing H . Problem 2.26 Find the centres of the following groups. (i) Integers Z, (ii) Dihedral group D4, (iii) Dihedral group D5, (iv) 2 × 2 General linear group GL2(Q), and (v) Permutation group S3. Problem 2.27 Prove that if H,J ≤ G, then o(H J )o(H ∩ J)= o(H)o(J). One method is as follows. Define a map θ : H × J → HJ by (h, j)θ = hj . Show that if g = hj where h ∈ H and j ∈ J , then gθ−1 = ha, a−1j : a ∈ H ∩ J , by proving inclusion both ways round. Further, show that if (ha, a−1j)= (hb, b−1j) then a = b, and so o(gθ−1) = o(H ∩ J). Lastly, count ordered pairs using the prop- erty o(H × J)= o(H)o(J). Note that (a) HJ need not be a subgroup of G, and (b) a second proof of this result is given in Theorem 5.8. Problem 2.28 Suppose G is a finite simple group of even order. Using Prob- lem 2.8, show that G is generated by its involutions. (Hint. Note that an involution is self-inverse.) By the Feit–Thompson Theorem (Chapters 11 and 12), this shows that all finite non-Abelian simple groups are generated by a set of their involutions. 40 2 Elementary Group Properties Problem 2.29 (Double Cosets) Suppose H,J ≤ G and a ∈ G.ThesetHaJ = {haj : h ∈ H,j ∈ J } is called the double coset of a with respect to H and J . Show that (i) Each element of G belongs to exactly one double coset. (ii) G is the disjoint union of its double cosets. (iii) Each double coset (with respect to H and J ) is a union of right cosets of H , and a union of left cosets of J . (iv) The number of right cosets of H in the double coset HaJ is [J : J ∩ a−1Ha]. Hence [G : H]= [J : c−1Hc] c∈C provided this sum is finite, where C is a set of double coset representatives for H and J . (v) Using the notation set up in Section 3.1,ifG = S4, a = (1, 2), H =(1, 2, 3), J1 =(1, 2, 3, 4) and J2 =(1, 4)(2, 3), write out the double cosets HaJi for i = 1, 2. ≤ ≤···≤ Problem 2.30 (i) Suppose J 1 J2 G, that is we have an infinite sequence = ∞ ≤ of subgroups of G.LetJ i=1 Ji . Show that J G. Note that in general a union of subgroups is not itself a subgroup. (ii) In (i), if Ji is simple for infinitely many i, show that J is also simple. Problem 2.31 (Project) Whilst reading this book, list all those theorems which ap- ply without restriction or caveat, for example, one of the first is Cancellation (The- orem 2.6). Chapter 3 Group Construction and Representation Bertrand Russell defined the integer ‘3’ as that property common to all sets having three elements, with similar definitions for other positive integers. The abstract en- tity ‘3’ has many representations in the myriad of sets with three elements.1 So it also is with groups. For example, the alternating group A5 to be introduced in Sec- tion 3.2 is defined as the group of even permutations on a five-element set—the first representation of this group we give is expressed in terms of permutations. But it has several other representations (as a matrix group or a symmetry group, et cetera, see Web Section 3.6). The point being that when we discuss an individual group, we almost always discuss a particular representation of the group, as a matrix, or permutation, or other type of group, and the corresponding ‘abstract’ group is that entity common to all of these representations—this is an important point to bear in mind when discussing individual groups. Also these ideas have led to a branch of the subject called “group representation theory” which we shall introduce in Web Chapter 13. There is one type of representation which comes close to the ‘abstract’ group, that is a ‘group presentation’ which was introduced briefly on page 21. The group is defined on an ‘alphabet’, the elements are the ‘words’ in this alphabet, and the oper- ation is defined using concatenation. For instance, one presentation of the dihedral group D3 described on page 3 is a,b a3 = b2 = (ab)2 = e . In this representation, the elements of D3 are given by the products of powers of a and b subject to the conditions given above. Assuming that we can apply the usual elementary group rules, it is easy to show that this system only has six members: e,a,a2,b,ab, and a2b, and that they ‘correspond’ to the six symmetries of the tri- angle described on page 3. We shall consider this topic in more detail in Section 3.4. 1Nowadays a more inductive definition is given using the integer zero and the successor function. H.E. Rose, A Course on Finite Groups, 41 Universitext, DOI 10.1007/978-1-84882-889-6_3, © Springer-Verlag London Limited 2009 42 3 Group Construction and Representation In this chapter, we discuss three methods for defining groups, that is, three types of group construction or representation. For another, see Section 12.1. They are de- fined using (a) permutations, (b) matrices, or (c) generators and relations, that is, presentations, and they all have important roles to play in the theory. We begin by describing the basic properties of permutations, and the symmetric and alternating groups. Historically, the development of the theory started with them and A. Cayley (1821Ð1895) in 1850 showed that every group can be represented as a permutation group (Theorem 4.7). And it is for this reason that some authors describe group the- ory as ‘the science of symmetry’; see Weyl (1952). Next we give the basic properties of matrix groups, many of the more ‘interesting’ groups in the theory, especially many simple groups, arise first as matrix groups; see Chapter 12. Lastly, we give the formal definition of a group presentation which will be further discussed and verified in Web Section 4.7. 3.1 Permutations We begin by developing the basic properties of permutations. Remember that we always read from left to right. Most of the work in this section first appeared in print in a series of papers published in the 1840s by the French mathematician A. Cauchy (1789Ð1857); see Section 6.1. The notation introduced below is also due to him. Definition 3.1 A permutation σ onasetX is a bijection of X to itself. As we shall normally be using finite sets X, it is convenient, but not essential, to take X ={1, 2,...,n} when o(X) = n. Apart from their natural ordering, no arithmetical properties of the integers 1 to n are used, they are just easily recognised labels for the elements of a set with n elements. We use two notations. First we have the ‘matrix’ form: 12... n σ = , (3.1) a1 a2 ... an where ai ∈ X and iσ = ai ,fori = 1,...,n; see the note on ‘left and right’ on page 68. Each element in the second row of (3.1) is the result of applying the per- mutation σ to the element in the first row directly above it: i → ai , i = 1,...,n, so no two elements in the second row are equal (some authors just print the second row taking our first row as read). The order of the columns is unimportant, but we usually write them as in (3.1). Using the fact that composition of two bijections is a bijection (Appendix A), we define the ‘product’ of two permutations by Definition 3.2 Let σ and τ be permutations on the same set X, where, for i ∈ X, iσ = ai and iτ = bi .Theproduct στ is given by = = ∈ i(στ) (iσ )τ bai for all i X. 3.1 Permutations 43 Using the matrix form (3.1) above, we can rewrite this as 12... n 12... n στ = a1 a2 ... an b1 b2 ... bn = 12... n a1 a2 ... an a1 a2 ... an ba1 ba2 ... ban 12... n = , ba1 ba2 ... ban where the second matrix in the second line is the same as the second matrix in the first line except that its columns have been permuted by σ . This does not affect the result. In the next section, we show that this product generates a number of new groups. The neutral element is the permutation that moves no element of X (the identity map ι on X), and the inverse of a permutation is its reverse, that is, if σ is given by (3.1), then − a a ... a σ 1 = 1 2 n . 12... n The second notation for permutations uses cycles. We begin with an example. Let 123456789 σ = . 1 731852694 This permutation maps 1 → 7, 7 → 6, 6 → 2, 2 → 3 and 3 → 1 so forming a cycle with five entries (1, 7, 6, 2, 3). Alternatively, we can write = 2 = 3 = 4 = 5 = 1σ1 7, 1σ1 6, 1σ1 2, 1σ1 3, and 1σ1 1. As the symbol 4 has not been used so far, we can start again: σ1 maps 4 → 8, 8 → 9 and 9 → 4, giving another cycle with three entries (4, 8, 9) in this case, which is disjoint from the first cycle. Finally, 5 is the only symbol in X which has not so far been used, and 5σ1 = 5 giving a third cycle with only one element. So we can treat σ1 as the ‘product’ of these three cycles, that is, we write σ1 = (1, 7, 6, 2, 3)(4, 8, 9)(5) (although sometimes single cycles, like (5), are taken as read and not printed). Note that the cycles are disjoint from one another, and the order in which they are written does not affect the final outcome. This example is typical as we show below. 44 3 Group Construction and Representation Definition 3.3 Let σ be a permutation on the finite set X and let x ∈ X. The ordered k-tuple x,xσ,xσ2,...,xσk−1 , where k is the smallest positive integer with the property x = xσk is called the cycle of length k,orthek-cycle, containing x. The integer k is called the length of the cycle.2 Some authors use the term ‘transposition’ for a 2-cycle, say (x, y),thatis,a permutation which maps x → y and y → x. Notes We use the same notation for an ordered k-tuple and a cycle, no confusion should arise. No two elements of a cycle are equal and 1 ≤ k ≤ o(X). Also compared with our first ‘matrix’ notation, cycle maps are horizontal left to right except that the last entry is mapped to the first, see footnote. The following result gives the essential facts about cycles. Theorem 3.4 Suppose σ is a permutation of the set X ={1, 2,...,n}. −1 (i) If τ is the cycle (a1,a2,...,an), then τ = (an,an−1,...,a1). (ii) The permutation σ can be expressed as a product of cycles τ1,τ2,...,τk, where k ≥ 1. They are disjoint and commute in pairs. (iii) The representation of σ given in (ii) is unique except for the order in which the cycles τi appear in the product. Proof (i) This follows immediately from the definition. (ii) We repeat the argument given in the example above. The sequence 1, 1σ, 1σ 2,... forms a cycle C1 of length k1, where k1 is the least positive integer satisfying k 1σ 1 = 1. If C1 = X, the result follows, for then σ forms a single cycle. If not, let a1 be the least positive integer in X not used in C1, and consider the 2 cycle C2 = (a1,a1σ,a1σ ,...) of length k2, where k2 is defined in a similar way to k1.Now C1 ∩ C2 =∅. (3.2) r s For if not, positive integers m and n exist satisfying 1σ = a1σ , which gives r−s a1 = 1σ contrary to our assumption that a1 ∈/ C1. We can continue this process forming C3,C4,... until all of X is used up, and then σ = C1C2 ···. By (3.2), each Ci is disjoint from the other cycles, and they commute for the same reason. (iii) The result clearly holds for 1-cycles. Suppose σ = τ1 ···τr = ν1 ···νs where τ1,...,τr (ν1,...,νs , respectively) are disjoint cycles each of which 2For ease of typesetting, cycles are printed linearly, but perhaps they should be printed in a circle as this would more truly represent them. 3.1 Permutations 45 moves at least two entries. Let 1 ≤ i ≤ n, and suppose both τk and νl move = = t = t = t i,soiτk iσ iνl . Then iτk iσ iνl for all t by assumption. Us- ing Problem 3.4(iii), this shows that τk = νl , and so τ1 ···τk−1τk+1 ···τr = ν1 ···νl−1νl+1 ···νs . We can repeat this argument, so use induction. Note that in this proof k1 + k2 +···=n = o(X). We need to prove the following two results before we can introduce the next topic—even and odd permutations. Lemma 3.5 Every permutation on a finite set can be expressed as a product of 2-cycles. Proof By Theorem 3.4, every permutation σ can be expressed as a product of cycles. Hence we need to show that every cycle equals a product of 2-cycles. But this follows by relabelling from the identity (1, 2,...,n)= (1, 2)(1, 3) ···(1,n), which the reader should check noting we read from left to right. We can extend this result to: Every permutation on X ={1, 2,...,n} can be ex- pressed in terms of the 2-cycles (1, 2), (1, 3),...,(1,n); see Problem 3.1. Cyclic structure is preserved by conjugation, see Definition 2.28.Wehave Theorem 3.6 Suppose σ and τ are permutations on X ={1, 2,...,n}. The permu- tations σ and τ are conjugate (a permutation α on X exists satisfying τ = α−1σα) if and only if σ and τ have the same cyclic structure, in which case τ can be obtained by applying α to the symbols of σ . First, we consider an example. Let X ={1,...,6}, and let σ = (1, 5)(3)(2, 6, 4), τ = (2, 3)(6)(4, 5, 1) and α = (1, 2, 4)(3, 6, 5). We have 2α−1 = 1, 1σ = 5 and 5α = 3, and so 2α−1σα = 3. This agrees with τ which also maps 2 to 3. Repeating this for the other members of X shows that α−1σα= τ .Alsoas1α = 2,...,6α = 5, we have (1α, 5α)(3α)(2α, 6α, 4α) = (2, 3)(6)(4, 5, 1) = τ, that is, if we replace the entries in σ by their images under α, we obtain τ . Con- versely, note that if we construct a ‘permutation matrix’ whose top row is the entries of σ , and whose bottom row is the entries of the cycle τ , we obtain α, viz.: 153264 α = , 236451 46 3 Group Construction and Representation which is a matrix form of α. This only works because σ and τ have the same cyclic structure—a product of a 2-cycle, a 1-cycle and a 3-cycle. Proof Suppose first σ and τ have the same cyclic structure with cycle lengths corresponding: σ = (...)···(...,l,m,...)···(...), τ = (. . .) ···(...,l ,m ,...)···(...). Form α (using its ‘matrix’ form, see (3.1)) by taking the entries of σ as its top row and the entries of τ as its bottom row: ... l m ... α = . (3.3) ... l m ... Now as lσ = m, lα = l so l α−1 = l, and mα = m ,wehave − mα = m so lσα = m , and so l α 1σα= m . But l τ = m , therefore we can deduce τ = α−1σα, that is, σ and τ are con- jugate, because the above calculation can be applied to all corresponding con- secutive pairs of entries in σ and τ (if l is the last entry in a cycle then m is the first; see footnote on page 44). For the converse, suppose σ has the form σ = (...)···(...,j,k,...)···(...), as above. Further, suppose jα = r and kα = s, then as jσ = k we have − kα = s so jσα = s, and so rα 1σα= s. Now as α and σ are permutations (bijections on X), so is α−1σα. Therefore, if we define τ by τ = α−1σα, the conjugate of σ by α, then it has the same cyclic structure as σ because the replacements l → r and m → s are themselves bijective, and the cyclic structure of σ is unaltered by this procedure. Returning to the example on page 45, we note that to obtain the given α we had to write σ and τ in the ‘right’ way. The cycles (4, 5, 1), (1, 4, 5) and (5, 1, 4) are identical, and so a number of different α can be constructed using the process given in the proof of the theorem. This is to be expected because there is likely to be a number of different solutions α to the equation ατ = σα. For example, we can use 153264 α = = (1, 2)(3, 6, 4, 5). 236145 3.1 Permutations 47 The reader should check that this new α also gives the correct answer, and write out the remaining possible solutions (Problem 3.2). Even and Odd Permutations A permutation is even (odd) if it can be expressed as a product of an even (odd, respectively) number of 2-cycles. This is not a definition as it stands because there are many ways of expressing a permutation as a product of 2-cycles (Lemma 3.5). For example, the permutation (5, 6)(3, 4) equals, amongst others, (1, 2)(4, 3)(5, 6)(2, 1), (2, 3)(2, 4)(2, 3)(6, 5) and (5, 1)(2, 3)(2, 4)(1, 6)(1, 5)(3, 2), and so they all represent the same permutation. But in each case the number of 2-cycles is even; this is typical of the general situation as we show now. Suppose o(X) = n and we are considering permutations on X. We introduce the following polynomial f which we use to ‘codify’ all possible 2-cycles. It is given by f(x1,...,xn) = (xj − xi). (3.4) 1≤i fσ(x1,...,xn) = f(x1σ ,...,xnσ ) = (xjσ − xiσ ). 1≤i Clearly, as σ is a permutation on X, each factor xj − xi of f in (3.4) occurs as a factor of fσ and vice versa, but the sign of this factor may be altered by σ . Hence, for all σ we have fσ =±f , and we use this fact to define the terms even and odd. Definition 3.7 (i) Using the notation set out above, we say that the permutation σ is even if fσ = f , and odd if fσ =−f . (ii) The sign,sgn(σ ), of the permutation σ is given by 1ifσ is even, sgn(σ ) = −1ifσ is odd. We show now that our two ‘definitions’ agree, and begin by proving Lemma 3.8 If the permutation σ is a single 2-cycle, then sgn(σ ) =−1, that is fσ =−f where f is given by (3.4). 48 3 Group Construction and Representation Proof Suppose σ is the 2-cycle (i, j) where 1 ≤ i (xk − xi)σ =−(xj − xk) and (xj − xk)σ =−(xk − xi). These sign changes occur in pairs and so when combined they have no overall effect, the result follows. Theorem 3.9 (i) If σ,τ are permutations on X, then sgn(σ τ) = sgn(σ ) sgn(τ). k (ii) If σ = τ1 ···τk and each τi is a 2-cycle, then sgn(σ ) = (−1) . Part (ii) shows that our two definitions of even and odd agree. Proof (i) For i ∈ X,wehave(iσ )τ = i(στ) by associativity of composition (Appendix A), hence f (σ τ)(x1,...,xn) = xj(στ) − xi(στ) 1≤i 3.2 Permutation Groups In the previous section, we studied properties of permutations, here we use these properties to construct a number of groups. Historically, these developments had a considerable influence on the progress of the theory as a whole; in the next chapter (Theorem 4.7), we show that every group can be treated as a group of permutations—a result due to Cayley. Note that there is nothing unique about per- mutations, in Problem 4.17 we show that every finite group can also be treated as a matrix group in many different ways; see also Web Section 4.7. First, we prove that the collection of all permutations on a set forms a group. Theorem 3.10 Suppose X is a set. The collection of all permutations on X, with composition as the operation, forms a group. 3.2 Permutation Groups 49 Proof A permutation is a bijection on X. The facts that composition of two bijections is a bijection, and composition of maps is associative, are proved in Appendix A. The neutral element is the permutation which moves no elements of X (the identity map ι, page 281), and the inverse of a permutation σ is its reverse: If a,b ∈ X and aσ = b, then bσ −1 = a; see page 43. This proves the theorem. The group of all permutations on X is denoted by SX.Ifo(X) = n (where we usually take X ={1, 2,...,n}), the group is denoted by Sn and called the symmetric group on n symbols. These groups have the following basic properties: (a) o(Sn) = n!. There are n! distinct bijections of an n-element set to itself. (b) If m ≤ n, then Sm ≤ Sn.InSn, consider all permutations which keep the same n − m elements fixed. For instance, S4 ≤ S5; in fact, S5 contains five copies of S4; see Problem 4.20(ii). (c) S1 = e , S2 is cyclic, and Sn is not Abelian if n>2; for instance, the cycles (1, 2) and (1, 2, 3) do not commute. (d) By Lemma 3.5, Sn is generated by its 2-cycles, it can also be generated by a two element set; see Problem 3.1. (e) By Theorem 3.6, two elements σ,τ ∈ Sn are conjugate in Sn (Definition 2.28)if and only if they have the same cyclic structure—a result we use several times. The subset (subgroup, see Theorem 3.11 below) of even permutations in Sn given by Definition 3.7 forms an important subgroup of Sn;forn>4, this provides our first example of a class of non-Abelian simple groups. We begin with Theorem 3.11 For n>1, the set of even permutations in Sn forms a normal sub- group of Sn with order n!/2. Proof A permutation is even if it can be expressed as an even number of 2-cycles (Theorem 3.9). Clearly, the identity permutation is even. Also the product of two permutations each a product of an even number of 2-cycles is itself a product of an even number of 2-cycles, and the inverse of a product of 2-cycles is the product written in reverse order. Hence, by Theorem 2.13, the set of even permutations forms a subgroup of Sn. It is normal by Theo- rems 2.29 and 3.6, as this last result shows that conjugation does not alter the cyclic structure of a permutation. Exactly half of all permutations in Sn are even, hence the order of the subgroup is n!/2. We prove this as follows. Define a map θ from the set of even permutations to the set of odd permutations by σθ = (1, 2)σ where σ is even. This is a bijection, for if τ is odd, then (1, 2)τ is even and (1, 2)τθ = (1, 2)(1, 2)τ = τ . Hence θ is surjective. A surjective map on a finite set onto itself is injective (Appendix A). The result follows. 50 3 Group Construction and Representation This last property extends to all subgroups of Sn, see the example on page 77 and Problem 3.7(ii). Alternating Groups The subgroup of even permutations of Sn given by Theorem 3.11 is called the alter- nating group on n symbols, it is denoted by An. Note that o(A1) = o(A2) = 1, and o(A3) = 3asA3 contains just two 3-cycles [(1, 2, 3) and (1, 3, 2)] and the identity permutation. A4 is the only non-Abelian alternating group that contains a (proper non-neutral) normal subgroup; see Problem 3.10 and page 158. 3 For n>4, the groups An are simple, and they form our second infinite collection of simple groups. We give an elementary proof of this simplicity result now; see comments at the end of the proof. The first step is to show that An is generated by its 3-cycles in the same way that Sn is generated by its 2-cycles (Lemma 3.5). Theorem 3.12 The group An is generated by its 3-cycles, provided n>2. Proof By Theorem 3.9, every element in An is equal to a product of an even number of 2-cycles. We prove the result by showing that all products of two 2-cycles are products of 3-cycles. Suppose 1 ≤ i, j, k, l ≤ n, and they are distinct. Then we have (i, j)(i, j) = e, (i, j)(i, k) = (i,j,k) and (i, j)(k, l) = (i,l,j)(j,k,l); the result follows. We come now to the main simplicity proof. We begin by assuming the contrary, that is, a proper subgroup K exists satisfying e =K An, then we show (a) if K contains a 3-cycle, then K contains all 3-cycles, and so by Theorem 3.12, K = An; and (b) K contains a 3-cycle if n>4. These two propositions prove the theorem because they show that An does not con- tain a proper non-neutral normal subgroup. First, we prove (a). Lemma 3.13 If K An and K contains a 3-cycle, then K = An. Proof As K An, K contains conjugates of all of its elements by Theo- rem 2.29. Hence, by Theorem 3.12, we need to show that 3-cycles are conju- gate in An. By Theorem 3.6, all 3-cycles are conjugate in Sn, but we cannot apply this directly because the conjugating element relating two 3-cycles may not belong to An; we must prove that all 3-cycles are conjugate by even ele- ments. We show this directly using elementary permutation arguments, later 3For n = 5 this result is effectively due to Abel, it was a corollary of his work on the non-solubility of the quintic. 3.2 Permutation Groups 51 (Problem 5.25) we give another proof of this result which ‘sheds more light’ on the underlying structure. We consider A5 first. Two distinct 3-cycles will overlap by one or two elements. Suppose there is a single overlap then, relabelling if necessary, we can take the 3-cycles in the form (1, 2, 3) and (1, 4, 5). We have (2, 4)(3, 5) ∈ A5 and (2, 4)(3, 5)(1, 2, 3)(3, 5)(2, 4) = (1, 4, 5). Similarly, if there is a double overlap, that is, the cycles are of the type (1, 2, 3) and (1, 2, 4),or(1, 2, 3) and (2, 1, 4), then (1, 4, 3), (3, 4, 5) ∈ A5 and (5, 4, 3)(1, 2, 3)(3, 4, 5) = (1, 2, 4) (3, 4, 1)(1, 2, 3)(1, 4, 3) = (2, 1, 4). The result follows for A5 by permuting the symbols 1,...,5. Secondly, we use this to prove the general result. If n>5, then An contains copies of A5 obtained by considering only those permutations that move the symbols of a fixed 5-element subset of {1,...,n}. In one of these copies of A5, moving the symbols 1, 2, 3,l,∗ only, we have (1, 2, 3) is conjugate to (1, 2,l), and so they are also conjugate in An. In a second copy, moving the symbols 1, 2,j,k,lonly, we have (1, 2,l)is conjugate to (j,k,l)in An. Hence (1, 2, 3) is conjugate to (j,k,l)in An, and the lemma is proved when n>4. (It is also true when n = 3 or 4, the reader should check this.) Theorem 3.14 If n>4, then the group An is simple. Proof We use a similar method to that given in the previous proof. Suppose K An and e =K. As noted above ((b) on page 50), we need to show that K contains a 3-cycle, the theorem then follows by Lemma 3.13.Letτ ( = e) be an element of K which moves the least number of symbols in the underlying set {1,...,n}. We show that τ is a 3-cycle. It cannot be a 2-cycle because 2-cycles are odd, hence if it is not a 3-cycle, it must move at least four symbols. It cannot be a 4-cycle because 4-cycles are odd. Hence it satisfies one of: (i) τ is a product of an even number of disjoint 2-cycles, (ii) τ is a product of a cycle of length at least 3 and further disjoint cycle(s). Therefore, τ is one of the following types τ1 or τ2, where σi is a cycle or a product of cycles whose entries are disjoint from those of its predecessors: (i) τ1 = (1, 2)(3, 4)σ1 (ii) τ2 = (1, 2, 3,...)(4, 5,...)σ2. 52 3 Group Construction and Representation In each case σi may be absent, and in (ii) the first cycle in τ2 has length at least 3, also the second cycle may have length 2 or may be longer, but we write (one of) the longest cycles first. We use τi to construct a new permutation which is not e and which moves fewer symbols than τi , this contradicts our supposition and so proves the theorem. In both cases, this new permutation is [ ]= −1 −1 = τi,α τi α τiα where α (3, 4, 5). We show [τi,α] moves fewer symbols than τi . Note that as τi ∈ K An,all conjugates of τi by elements of An belong to K, and so [τi,α]∈K. (i) Applying the permutations that make up [τ1,α] in turn, we have: beginning with the base set 1 2 3 4 5 ..., −1 ∗ applying τ1 we obtain 2 1 4 3 ..., applying α−1 we obtain 2 1 3 5 ∗ ..., applying τ1 we obtain 1 2 4 ∗∗..., applying α we obtain 1 2 5 ∗∗.... So if we apply [τ1,α] to our base set we obtain a permutation which fixes 1 and 2, and maps 3 to 5, that is, we have in K a non-neutral permutation which moves fewer symbols than τ1, contradicting our assumption. (ii) If we apply the permutation [τ2,α] to our base set {1, 2, 3, 4, 5,...} we obtain a permutation which fixes 2 and maps 3 to 4, and so again we have a contradiction; the reader should check this. In both cases, we have constructed in K a non-neutral permutation moving fewer symbols than those moved by τ . Therefore, our assumption is false, τ is a 3-cycle, and the theorem follows. There are many proofs of this result in the literature. The proof given above has the advantage that it uses a minimal amount of ‘apparatus’, and so it can be presented here. Further proofs of this result are given in Problems 3.16 (using con- jugation), 5.25 (using centralisers), and 6.16 (using the Sylow theory) and in Web Sections 3.6 and 14.1. Suzuki (1982, page 295) gives a proof that uses Bertrand’s 2 postulate! He shows that o(An) = o(K) if K An, and then he applies the postulate (which states that for all positive integers m aprimep can be found lying between m and 2m). Up to isomorphism, A5 is the only non-Abelian simple group with order less than 168, see Problems 6.15 and 6.17, and Chapter 12. 3.3 Matrix Groups Matrix algebra provides a wide range of group examples. Given a field F and a positive integer n, we consider the set of all n × n non-singular matrices with entries in F .(Ann × n matrix A is non-singular if and only if another n × n matrix B 3.3 Matrix Groups 53 exists satisfying AB = BA = In. A standard theorem of linear algebra states that A is non-singular if and only if det A = 0 where det A denotes the determinant of A.) This set of matrices forms a group called the general linear group over F , and it is denoted by GLn(F ). The neutral element is the n × n identity matrix In, and inverses exist by definition. The normal subgroup (Theorem 3.15 below) of matrices with determinant 1 is denoted by SLn(F ) and called the n × n special linear group over F . We shall mainly be concerned with the case when F is finite, see Section 12.2. Up to isomorphism, unique finite fields exist of order pm for each prime number p and positive integer m; they are usually called Galois fields, and denoted by Fpm , after their discoverer É. Galois;4 see page 229. The general linear group defined over m m a field with p elements, that is, GLn(Fpm ), is denoted by GLn(q) where q = p . Similarly, we write SLn(q) for SLn(Fpm ). The basic properties of these groups are given by Theorem 3.15 Suppose F is a field and n ≥ 1. (i) The set of matrices GLn(F ) with matrix multiplication forms a group. (ii) SLn(F ) GLn(F ). (iiia) If q = pm, where p is prime and m>0, then n(n−1)/2 n n−1 o GLn(q) = q q − 1 q − 1 ···(q − 1). (iiib) If n>1, o(SLn(q)) = o(GLn(q))/(q − 1), and o(SL1(q)) = 1. Proof (i) and (ii) Both of these follow from the fact that det AB = det A det B −1 for all A,B ∈ GLn(F ), and its corollaries det A = 1/ det A, and det In = 1. (iiia) Let A ∈ GLn(q).AsA is non-singular, every sequence of n elements n of Fq is a possible top row of A except {0, 0,...,0}, that is, there are q − 1 possible top rows for A.AgainasA is non-singular, a possible second row of A is a sequence of n elements of Fq which is not a linear multiple of the first row. There are q multiples of the first row (this includes the row {0, 0,...,0}), so qn − q = q(qn−1 − 1) second rows of A are possible. We can continue this process, the third row must not be a linear combination of the first two rows, and so on. The result follows by collecting terms. (iiib) This is a consequence of (ii) and (iiia) as the number of non-zero ele- ments of Fq is q −1, see page 76, or we can use the following argument. In the last stage of the procedure given in (iiia), there are qn − qn−1 = qn−1(q − 1) choices for the last row of the matrix. So there are qn−1 choices if we also stipulate that the determinant of the matrix in question has a particular value. This gives (iiib) if we choose this value to be 1; the reader should check this. For example, we have o(GL3(2)) = 168 and o(SL2(3)) = 24. 4The fact that all finite fields have this form was first proved by E.H. Moore in 1893. 54 3 Group Construction and Representation The groups SLn(q) give rise to our third class of simple groups. If SLn(q) is ‘factored’ by its centre then the resulting linear group which is denoted by Ln(q) is simple, except when n = 2 and q = 2 or 3. The operation of forming a factor group will be discussed in the next chapter, and a simplicity proof will be given in Chapter 12. This result implies that SLn(q) is ‘nearly’ simple, its only (proper non- neutral) normal subgroup being its centre which has order (q − 1,n). In some cases, SLn(q) is itself simple (when its centre has order 1), examples are SL2(4) (which is isomorphic to A5, see Problem 6.16) and SL3(3) (Section 12.2). Subgroups of GLn(q) The group GLn(q) has a number of important subgroups, one is SLn(q) as dis- cussed above. Another type is the subgroup of permutation matrices discussed in Problem 3.12. A third type is the subgroup of ‘upper triangular’ matrices. (For a full list, see Dickson 2003, or Kleidman and Liebeck 1990.) A matrix A ∈ GLn(q) is called upper triangular if every entry in A below the main diagonal is zero, this subgroup is denoted by UTn(q). Lower triangular matrices can also be introduced. Note that each diagonal entry of a non-singular upper triangular matrix is non-zero because the determinant of this matrix equals the product of its diagonal elements. We have Lemma 3.16 UTn(q) ≤ GLn(q). Proof Clearly, In ∈ UTn(q), and the product of two upper triangular matrices is itself upper triangular. Hence, by Theorem 2.13, we need to show that the inverses are also upper triangular. Let A = (aij ) ∈ UTn(q) (where aij is the (i, j)th entry in the matrix A) and so aij = 0ifj sij = aiibij +···+ainbnj , with n − (i − 1) summands as ai1 = ··· = ai(i−1) = 0. If j s(n−1)j = a(n−1)(n−1)b(n−1)j + a(n−1)nbnj = 0. This gives b(n−1)j = 0 when j Suppose A ∈ GLn(q). We can find U1,U2 ∈ UTn(q) and a permutation matrix P (Problem 3.12) to satisfy A = U1PU2. This result is known as the Bruhat Decomposition Theorem, and it has a number of useful applications. We shall give a simple proof in the case n = 2, the proof of the 3.4 Group Presentation 55 = ab = general case follows similar lines. Let A cd .Ifc 0, then the result follows by putting U1 = A and P = U2 = I2,the2 × 2 identity matrix. So we may assume c = P = 01 U = x1 y1 U = x2 y2 that 0. In this case, we let 10 , 1 01 and 2 01 . Then, if U1PU2 = A we have x y 01 x y y x y y + x ab 1 1 2 2 = 1 2 1 2 1 = . 01 10 01 x2 y2 cd −1 −1 This gives x2 = c = 0, y2 = d, y1 = ac and x1 = b − dac = 0, as det A = 0. Now this defines both U1 and U2. × transvection 1 r 10 r = A2 2 is a matrix of the form 01 or r 1 where 0; we shall consider these matrices in more detail in Chapter 12. Here we use them to derive a consequence of Bruhat’s result as follows. Theorem 3.17 The group GL2(q) is generated by its diagonal matrices and its transvections. Proof By Bruhat’s Theorem we need to show that we can construct the up- per triangular and permutation matrices as products of diagonal matrices and transvections. We have a 0 1 ba−1 ab = , 0 c 01 0 c 11 10 01 = , and 01 −11 −11 01 −11 01 = , −11 01 10 which proves the result as there are only two permutation matrices in the 2-dimensional case. 3.4 Group Presentation The American mathematician W. von Dyck (1856Ð1934) in about 1880 introduced the ‘presentation’ of a group, as an example he derived the presentation of S4 given in Problem 3.18. This work has led to a branch of the theory called Combinatorial Group Theory, we shall only give the basic ideas and definitions, the interested reader should consult Lyndon and Schupp (1976) or Chandler and Magnus (1982). In the second section of Chapter 2, we introduced briefly a method for defining groups using so-called generators and relations, here and in Web Section 4.7 we give the formal definitions and proofs. The method is important in the theory because many groups are best defined in these terms, see, for example, the group E discussed in Section 8.3. Although it is a representation, it is the nearest ‘approach’ to an abstract definition of the group. 56 3 Group Construction and Representation Suppose we are given an alphabet or list A of letters or symbols which we usually assume to be finite (but this is not essential): a1,a2,...,b1,...,c1,.... A word is defined as a finite sequence of letters written using concatenation, for example, b2,c3c3b1a2c1, and a1a1a1a1 are words. In this section, we use lower case characters at the beginning of the al- phabet a,b,c,... for letters, and lower case characters at the end of the alphabet x,y,z,... for words.Ifthewordx has the form x = y1zy2, then z is called a sub- word of x (as are both y1 and y2). The operation is as suggested above, that is, concatenation. For example, if x = aiaj bk and y = bkclc1am, then xy = aiaj bkbkclc1am. It follows immediately that the system of all words on a fixed alphabet A with the operation of concatenation forms a semigroup. To construct a group we need to bring the neutral element and inverses into the system, and we do this as follows. The word containing no letters from A will act as the neutral element. Mainly for typographical reasons, it is necessary to introduce a symbol for this element, and so as previously we let e stand for it where e/∈ A.Note that the symbol e and the blank symbol are synonymous. For the inverse operation to apply, the alphabets have the form A ∪ A , where A ∩ A =∅and there exists a bijection between A and A . So the alphabet has the structure: a1,a1,a2,a2,...,b1,b1,...,c1,c1,..., and the given prime bijection between A and A satisfies (a ) = a, for all letters a ∈ A and a ∈ A . The basic idea here is that we want a to act as the inverse of a, et cetera. We define the set of reduced words on A by Definition 3.18 Awordx on the alphabet A (it is assumed that the second alphabet A is also present; see above) is called a reduced word if (a) it contains no pairs of consecutive letters (symbols) of the form aiai,aiai,bj bj ,..., that is, it contains no letter, with or without a prime , immediately followed by its image under the -map, and (b) all blank symbols have been removed except that if no alphabet letters remain, then a single blank symbol should remain. For example, c3,a1b2,a1a1a1,e are reduced, and a3a3,b2e,b1c1c1,a2a2a2a2 are not reduced. 3.4 Group Presentation 57 We now adapt the semigroup concatenation operation to construct a group opera- tion. Suppose x and y are reduced words. To form their ‘product’, which we call the reduced concatenation of x and y, we first construct the semigroup concatenation xy of x and y, and then we remove (that is, replace by the empty word) all pairs of consecutive symbols of the form aiai ,oraiai ,orbj bj , et cetera, that are formed = by the concatenation, or by previous removals. For example, if x a1a1b2c1 and = y c1b2a1, then = = = = ; xy a1a1b2c1c1b2a1 a1a1b2b2a1 a1a1a1 a1 = = and yx c1b2a1a1a1b2c1 c1b2a1b2c1. (This construction is a generalisation of one that will be familiar to the reader. In the group of integers Z, we have, for instance, 3 = 4 − 1 = 5 − 2 =···=(n + 3) − n. Note (n + 3) − n = 3 + (n − n) = 3 for all n ∈ Z, that is, each time we encounter n − n we replace it by 0, and delete it if other symbols are present. This is exactly mirrored in the general case described above.) This system forms a group. Two words are equal if each can be obtained from the other by the insertion and/or deletion of pairs of consecutive symbols of the form aa or a a. To be more precise this procedure defines an equivalence relation on the set of words, and the group elements are the corresponding equivalence classes, see Web Section 4.7 (In our example using the group Z given above, we associate 3 with the set of differences {(n + 3) − n : n ∈ Z}.) The set of words has a neu- tral element and is closed under the inverse operation, hence we need to establish associativity. We have Theorem 3.19 The system of reduced words on a fixed alphabet A (that is, A ∪ A , see page 56), with the operation of reduced concatenation defined above, forms a group. We shall not give a proof of this result here. This is best done once we have proved the First Isomorphism Theorem which is given in Chapter 4;seeWeb Sec- tion 4.7. It is also possible to give an ad hoc proof which splits the result into a number of particular cases, but it is somewhat unsatisfactory in that it does not il- lustrate the underlying structure, and it is not easy to be sure that all cases have been considered. Reader, try it. The group defined by the above theorem is called the free group on the alpha- bet A. It is called free because there are no constraints on the words in the group other than those needed to form the group. It is necessarily infinite, for if a ∈ A then a,aa,aaa,... all belong to the group, and they are distinct. A range of new groups can be defined by introducing more constraints. Consider the following ex- ample. Let A ={a}, that is, the alphabet consists of the two letters a and a where aa = a a = e, the empty word, and let G be the free group on A. Then the (reduced) elements are ...,a a ,a ,e,a,aa,aaa,..., 58 3 Group Construction and Representation and G is called the infinite cyclic group, it is denoted by Z. We can introduce a relation in the form an = e where n is a positive integer and an stands for aa ···a with n copies of a. It is easily seen that the elements of the new structure are e,a,a2,...,an−1, and that it forms a group with n elements. It is called the cyclic group of order n and is denoted by Cn. We can treat the elements of Cn in the same way as those of the infinite cyclic group except each time we encounter an we delete it (replace it bytheemptyword)aswedoforaa and a a. Further cyclic group properties are given in Section 4.3. The procedure illustrated in this example can be applied to all free groups. Sup- pose A is an alphabet and R is a set of words on A (that is, on A ∪ A ), then we can form the structure (group) H called a presentation. It is denoted by H = A | R , (3.5) and consists of the free group on A with the constraints (relations) x = e, for all x ∈ R.IfR ={x1,...,xk}, we usually write x1 =···=xk = e for R in this presentation. The elements of A are called the generators of H ,forx ∈ R the equation x = e is called a relation of H , and this method of defining H is called a presentation of H . (Some authors use the term relator for a word x in R.) To study H we work in the free group on A, and each time we encounter a word in R we replace it by the empty word just as we do for aiai,aiai,... in the free group. In the example n n above, we have Cn = a | a = e and a is replaced by the empty word each time it occurs. For all A and R the structure H forms a group, we shall prove this in Web Section 4.7 using the Isomorphism Theorems. We show that H can be defined as a ‘factor group’ of the free group on A. We also show that all groups can be treated in a similar manner. It can be difficult, and in some cases impossible, to determine the properties of H (see (3.5)), its order, or even if it is finite or infinite. For a general discussion of this and related considerations, including the Word Problem for Groups, see Rotman (1994). One difficulty is of the following type: Given a collection of relations R and a generator a, we might be able to deduce both am = e and an = e where (m, n) = 1, this would give a = e (use the Euclidean algorithm) and so a would be redundant. In the ‘worst’ case, the whole construction could collapse to the neutral group. To avoid this, and similar problems, in most cases it is necessary to find another repre- sentation of the group in question, as a matrix, permutation, or similar group, and to construct a bijection between the corresponding elements. We illustrate this in the following two examples. Todd and Coxeter have devised a method called coset enumeration which can be used to determine the structure of a group given by a presentation; see Coxeter and Moser (1984), Chapter 2. n 2 2 Example 1 (Dihedral group Dn)LetA1 ={a,b} and R1 ={a ,b ,(ab) }. Then n 2 a,b | a = b = abab = e gives a presentation of the dihedral group Dn;see 3.5 Problems 59 Section 2.2. The order is 2n because it can easily be shown that every element of the free group on {a,b} can be reduced to a member of the set e,a,a2,...,an−1,b,ab,...,an−1b using the relations in R1. There is no collapse, see above, because our first repre- sentation of this group was as the symmetry group of the regular polygon with n sides and 2n symmetries. The group can also be generated by two involutions, see Problem 3.20. n −2 −1 Example 2 (Dicyclic Group Qn) Again let A1={a,b}, and let R2={a b ,abab }, n 2 2 and so Qn = a,b | a = b = (ab) . We show first that the relation a2n = e (3.6) is a consequence of the relations in R2; note we are assuming that the structure a,b | R2 is a group, this is proved in Web Section 4.7. Hence we can apply n 2 basic group properties. Assume first that n is even. The relations R2 give a = b and b = aba, and so we have, replacing b by aba several times: an = b2 = (aba)(aba) = a2ba4ba2 =···=an/2banban/2 = an/2b4an/2 = a3n, and so in this case (3.6) follows by cancellation; we leave it as an exercise for the reader to derive the remaining case. Now as in the first example we can show that a member of the group is of the form at bu where 0 ≤ t<2n and 0 ≤ u ≤ 1, and so the group has order 4n. Matrix representations of these groups are given in Prob- lems 3.13 and 3.22.Weusethetermquaternion group for the dicyclic group Q2, see Section 6.1. Also the term generalised quaternion group is used for the group (n+2) Q2n (with order 2 ) when n = 2, 3,...; see Problem 3.22. 3.5 Problems Problem 3.1 Show that Sn can be generated by each of the following sets: (i) {(1, 2), (1, 3), . . . , (1,n)}; (ii) {(1, 2), (2, 3),...,(n− 1,n)}; (iii) {(1, 2), (1, 2,...,n)}. As every finite group G can be treated as a subgroup of Sm for some suitably chosen m where m ≤ o(G) (Cayley’s Theorem, page 72), (iii) shows that every finite group is a subgroup of a group with two generators, or to put this another way, increasing the number of generators (above two) does not necessarily increase the complexity of the group. Also all finite non-cyclic simple groups have two element generating sets; see Cameron (1999). (iv) Show that An,forn>4, is generated by its involutions. Note that this property holds for all non-Abelian simple groups (Problem 2.28). 60 3 Group Construction and Representation Problem 3.2 If σ = (1, 2, 3)(4, 5, 6) and τ = (1, 5, 6)(2, 3, 4), find all α such that α−1σα= τ ; see the example given after Theorem 3.6. Problem 3.3 List the conjugacy classes of (i) S3, (ii) S4, (iii) S5,(iv)A3,(v)A4, and (vi) A5. Note that (v) and (vi) are not straightforward, some permutation cal- culations are needed; for a more detailed explanation, see the subsection on the Class Equations in Chapter 5. Secondly, find the normal subgroups of (vii) S4 and (viii) A4. Problem 3.4 Let σ,τ and ν be cycles in Sn. (i) Show that if σ and τ are disjoint and στ = e, then σ = e = τ . (ii) If σ commutes with τ and τ commutes with ν, does it follow that σ commutes with ν? (iii) Suppose σ and τ belong to SX where X ={1,...,n} and both σ and τ move i ∈ X. Prove that if iσr = iτr for all r ≥ 0, then σ = τ . (iv) Let p be a prime. Show that if σ p = ι, the identity permutation, then σ = ι or σ is a product (with possibly only one factor) of p-cycles. Problem 3.5 Give formulas for the orders of the elements of (i) Sn, and (ii) An, and list them when n = 7. Problem 3.6 Let p and q be primes where p | q − 1, let σ = (1, 2,...,q), and let 12... q τ = , where 0 Problem 3.7 (i) Show that Sn isomorphic to a subgroup of An+2. (ii) Using Theorem 3.11 and its extension given in the example on page 77,show that if J is a simple subgroup of the symmetric group Sn, then it is also a (simple) subgroup of the alternating group An. Problem 3.8 (Semi-regularity) A permutation σ on X ={1,...,n} is called semi- regular if (a) every element of X is moved by σ , and (b) σ can be expressed as a product of disjoint cycles all of the same length. The identity permutation is also called semi-regular (it is a product of n cycles each of length 1). (A permutation is called regular if it is semi-regular and transitive.) (i) Prove that σ is semi-regular if and only if it is a power of an n-cycle. (Hint, if σ = (a1,...,ar )(b1,...,br ) ···(d1,...,dr ), consider the cycle θ = (a1,b1,...,d1,a2,b2,...,d2,a3,...,dr ).) 3.5 Problems 61 (ii) If τ is an n-cycle, show that τ s is a product of (n, s) disjoint cycles each of length n/(n, s) (note that (n, s) denotes the GCD of n and s). Deduce that if p is a prime, then each positive power of a p-cycle is either a p-cycle or the identity permutation. Problem 3.9 (Maximal Subgroups of Sn) (i) Show that Sk ≤ Sn,for1≤ k ≤ n, and determine a lower bound on the number of copies of Sk that occur in Sn; see also Problem 4.20(ii). (ii) For 1 ≤ k ≤ n,letSk × Sn−k denote the set of elements in Sn with the form of a permutation using the symbols in {1, 2,...,k}=Y only, multiplied by a permuta- tion using the symbols in {k + 1,k+ 2,...,n}=Z only. (This is the direct product of Sk and Sn−k, see Chapter 7—you need to check that Sk Sk × Sn−k, et cetera) Show that Sk × Sn−k forms a subgroup of Sn. If k = n − k, this subgroup is maximal in Sn. This can be proved as follows: Show that the subgroup generated by the elements of Sk,Sn−k and a 2-cycle (u, v) where u ∈ Y and v ∈ Z is the whole of Sn; you are not asked to prove this here, but you could try it, say when n = 5 or 6. Note that the same general argument applies if we replace the first set Y by an arbitrary k-element subset of {1, 2,...,n}=N provided the second set Z is replaced by its complement in N. (iii) By (ii) we have Sn × Sn is a subgroup of S2n; show that it is not maximal 2 by constructing a new subgroup of S2n of order 2(n!) . (Hint. Begin with Sn × Sn, add an element of order 2, and then show that the new set forms a proper subgroup of S2n.) As in (ii) the new subgroup constructed here is maximal in S2n. The properties described in (ii) and (iii) form part of the O’Nan–Scott Theorem which provides a complete description of the maximal subgroups of the symmet- ric groups. A number of these subgroups involve so-called wreath products which are introduced on page 156. For further details, the reader should consult Cameron (1999), page 107. Problem 3.10 (Alternating Group A4) There are four main ways to represent A4 as follows: (a) The group of even permutations on the set {1, 2, 3, 4}, (b) The group with presentation a,b | a2 = b3 = (ab)3 = e , (c) The symmetry group of a regular tetrahedron (with four equilateral triangular sides), and (d) SL2(3) ‘factored by its centre’; see Chapters 4, 8 and 12. (i) Using direct calculation show that (a), (b) and (c) define isomorphic groups, see Problem 4.4(iv) for (d). (ii) Find the subgroups of A4, and indicate which are normal. 62 3 Group Construction and Representation Problem 3.11 (i) In this problem you are asked to construct a subgroup J of S5 with order 20. Let σ = (1, 2, 3, 4, 5). Choose a 4-cycle τ so that the subgroup J = σ,τ has this order. One method is as follows. Show that the group H = a,b | a5 = b4 = e, ba2 = ab has order 20, then find an isomorphism between H and J .It can be shown that J is a maximal subgroup of S5, it is usually called metacyclic and denoted by F5,4, see Theorem 6.18 and Web Section 6.5. Can a similar construction be undertaken in S7? (ii) Using Lagrange’s Theorem (Theorem 2.27) and Problem 2.20 find the sub- groups of A5 of order less than 10. It does also have proper subgroups of order 10 and 12, but none larger; see page 101. Problem 3.12 (Permutation Matrices) Permutation theory can be developed as a part of matrix algebra as follows. Given a permutation σ ∈ Sn,then × n per- mutation matrix Pn is formed from the n × n identity matrix In by permuting its rows by σ , that is, the first row (1, 0,...,0) becomes the 1σ th row, the second row (0, 1, 0,...,0) becomes the 2σ th row, and so on. Prove the following properties: (i) Each row and each column of Pn contains a single ‘1’ and n − 1 zeros. (ii) The determinant of Pn, det Pn, equals ±1. (iii) The inverse of a permutation matrix is a permutation matrix, and so the set of all n × n permutation matrices forms a subgroup of GLn(F ) for all fields F .Is this subgroup normal? (iv) A permutation σ is even if and only if the determinant of the corresponding permutation matrix is positive. Note that if the definition of the determinant function Δ uses the notions of even and odd permutations to determine the signs in the basic sums, then (iv) cannot be applied to define these permutations, but it can be so applied if Δ is defined as a multilinear alternating function which takes the value 1 on the identity matrix; see, for example, Rose (2002), Chapter 4. πi/3 η 0 Problem 3.13 Let η = e and C = − . 0 η 1 (i) Calculate C3,C6 and C−1. (ii) Choose a 2 × 2matrixD to satisfy CD = DC−1 and D2 = C3. (iii) Show that the group generated by C and D subject to the conditions in (ii) has order12, and gives a representation of the dicyclic group Q3. This construction can easily be extended to give representations of the groups Q2n+1 for all positive integers n. Problem 3.14 Show that GL2(4) is isomorphic to a subgroup of GL4(2) as fol- lows. First, note that the set of non-zero elements of a field of four elements forms a multiplicative cyclic group of order 3. Second, use this fact to show that GL1(4) is isomorphic to a subgroup of GL2(2). Now repeat this procedure in the case un- der consideration using block multiplication of matrices. A fair knowledge of linear algebra is needed to complete this problem. 3.5 Problems 63 Problem 3.15 Suppose F is a field. Let ITn(F ) be the subset of UTn(F ) (the group of n × n upper triangular matrices, see page 54) of those matrices with 1 at each main diagonal entry—they are sometimes called ‘unipotent’, and let IZTn,r (F ) be the subset of ITn(F ) of those matrices whose ith superdiagonals consist entirely of zeros for i = 1,...,r. We shall return to this example in Chapter 10 when dis- cussing nilpotent groups. Show that (i) ITn(F ) UTn(F ). (ii) IZTn,r (F ) ITn(F ), and IZTn,r (F ) IZTn,r−1(F ) if r = 1,...,n− 1. (iii) If o(F) = p (and so we work modulo p) show that ITn(F ) is a Sylow r r p-subgroup of GLn(F ), that is o(ITn(F )) = p where p is the largest power of p dividing o(GLn(F )); see Section 6.2. (iv) If A ∈ IZTn,r (F ) and B ∈ ITn(F ), then [A,B]∈IZTn,r+1(F ). Problem 3.16 Use Problem 3.3(vi) and Theorem 2.29(ii) to give another proof of the simplicity of A5. You should begin by noting that the neutral element e belongs to all subgroups. Problem 3.17 Show that the group Q does not have a finite generating set. 4 3 2 Problem 3.18 Prove that S4 has the presentation a,b | a = b = (ab) = e . Show also that if the powers 4, 3 and 2 are permuted, then further presentations of S4 are given. Problem 3.19 (i) Show that the group SL2(p) possesses only one involution when p is an odd prime. Also note Problem 12.5. (ii) Using a suitable computer program (or working by hand) show that the con- jugacy classes (Definition 2.11(iii)) of the group SL2(5) have orders (and number of classes) 1(2), 12(4), 20(2) and 30, with nine classes in all. (iii) Use (ii) to show that the only proper non-neutral normal subgroup of SL2(5) is isomorphic to C2. (iv) The maximal subgroups of SL2(5) are isomorphic to SL2(3), Q5 and Q3 with orders 24, 20 and 12, respectively. Find subgroups in SL2(5) isomorphic to these groups. The group SL2(5) has a special connection with so-called Frobenius comple- ments, this will be discussed in Web Section 14.3. Problem 3.20 Let D = c,d | c2 = d2 = e , A = c , and K = cd . (i) Show that A ≤ D, K D, A ∩ K = e , and AK = D if o(D) > 2. (ii) Use this to give new presentations of the dihedral groups Dn. Proposition (i) shows that D is isomorphic to a semi-direct product of K by A; see Section 7.3. As noted earlier (page 26) involutions (that is, elements of order 2) play an important role in the theory, especially in simple group theory. This also applies to groups of the form D generated by two involutions; see, for example, Aschbacher (1986), page 242. Note also that the jump from two to three generators 64 3 Group Construction and Representation can have dramatic effect, for very large and complex groups exist which have a generating set with just three involutions—for example, L3(3) with order 5616, see pages 264 and 265, and Problem 12.10. Problems 2.28, 3.21 and 3.23 should also be consulted. Problem 3.21 (A Presentation of Sn+1 for n>1) Let a1,...,an be symbols satis- fying the following conditions, where Qj,k does not apply if n = 2, P : 2 = ≤ ≤ i ai e for 1 i n, 2 Qj,k : (aj ak) = e for 1 ≤ k (i) Show that (a) aj and ak commute if 1 ≤ k Let G be the group generated by the set {a1,...,an} with the relations (∗), let H be the subgroup generated by the subset {a1,...,an−1}, and let Z be the set of cosets Z ={H,Han,Hanan−1,...,Hanan−1 ...a1}. (ii) Using (i), show that if Hy ∈ Z and 1 ≤ i ≤ n, then Hyai ∈ Z. There are a number of cases to consider. (iii) Use (ii) to show that [G : H]≤n + 1, and so by induction deduce o(G) ≤ (n + 1)!. (iv) By considering maps from G to Sn+1 of the form ai → (i, i + 1), show that G is a presentation of Sn+1 using Problem 3.1 and (iii). When Sn+1 is represented in this way, that is, when it is generated by a set of invo- lutions, it is known as a Coxeter Group. The theory of these groups has developed considerably in the past 30 years; see, for example, Suzuki (1982) and Björner and Brenti (2005). Problem 3.22 Suppose m is a positive even integer. (i) Let Rm be the subgroup of GL2(C) generated by 0 ω 01 A = and B = , m ω 0 −10 where ω is a primitive 2mth root of unity. Prove Rm Qm; see page 59. (ii) Prove that Qm has a unique involution C, and Z(Qm) = C . (iii) Further, show that Qm/Z(Qm) Dm. (Note that factor groups are discussed in Chapter 4.) If G is a 2-group (one whose order is a power of 2, see Chapter 6) and it has a single subgroup of order 2, then it has been shown that G is either cyclic or dicyclic; see Kurzweil and Stellmacher (2004), page 114. 3.5 Problems 65 Problem 3.23 Working in S8, investigate the subgroup F generated by the permu- tations a = (3, 4)(5, 6), b = (1, 3)(2, 4)(5, 7)(6, 8) and c = (1, 5)(2, 6)(3, 8)(4, 7). Construct a list of the elements with their orders, and so calculate o(F). Secondly, list the subgroups of F , and determine their orders, their normality status, and which of them is the centre. Finally, find a presentation for F , that is, find a collection of properties of a,b and c from which all remaining properties can be derived. The calculations can be done by hand, or by using a computer algebra package which includes the basic permutation operations. See also Problem 8.12. Problem 3.24 (Project—The Group GL2(3)) A number of groups play a special role in the theory, A5 being one of these. Another is the general linear group GL2(3), later (in Chapter 12) we shall describe some of its properties. Here you are asked to give representations in terms of the three main construction methods discussed in this chapter. In particular, (a) find three 2 × 2 matrices with orders 2, 3 and 8, respectively, which generate the group, (b) find three permutations of the set {1, 2, 3, 4, 5, 6, 7, 8} which generate the group as a subgroup of S8, and (c) show that the group has a presentation in the form a,b,c | a8 = b2 = c3 = e,bab = a3,bcb= c2,c2a2c = ab,c2abc = aba2 . You will need to find inclusion maps between these systems, and then show that each contains 48 elements (Theorem 3.15). Note that there is no unique answer to this problem, and that it can be done by hand, but a suitable computer algebra package would help with the calculations. This project will be continued in Problem 6.23. Chapter 4 Homomorphisms During the past half century and more, one of the underlying ‘themes’ in mathe- matics has been the realisation that maps are equally as important as objects or sets, particularly those that preserve or transfer properties from one object or system to another. They have been called natural maps, morphisms, or sometimes structure- preserving maps. We use the first of these names as a general term for these maps. Therefore, as we have introduced groups, our basic objects of study, we must now discuss the natural maps between them. In group theory, they are called homomor- phisms, and isomorphisms when the correspondence is ‘exact’; isomorphisms were introduced informally on page 17 in Section 2.1. Two groups can appear to be distinct, but are, in fact, identical from the group- theoretical point of view. For example, consider D3, the dihedral group of the tri- angle defined on page 3, and S3, the group of all permutations on the set {1, 2, 3}. They have identical group-theoretic properties, and a bijection θ between them that satisfies ghθ = gθhθ (4.1) articulates this fact; equation (4.1) is called the homomorphism equation.Themap θ : D3 → S3 can be defined as follows (it is not the only possible map; see Sec- tion 4.4). Referring to the definitions given on page 3, the rotation α ∈ D3 about the centre of the triangle by angle 2π/3 clockwise is mapped to (1, 2, 3), that is, we set αθ = (1, 2, 3).Using(4.1)wehave α2θ = αθαθ = (1, 2, 3)2 = (1, 3, 2), and ιθ = α3θ = (αφ)3 = (1, 2, 3)3 = e where ι denotes the ‘identity map’, see page 281. Also, the reflection β ∈ D3 about a vertical axis is mapped to (2, 3), that is, we set βθ = (2, 3). Then by (4.1)again H.E. Rose, A Course on Finite Groups, 67 Universitext, DOI 10.1007/978-1-84882-889-6_4, © Springer-Verlag London Limited 2009 68 4 Homomorphisms we have αβθ = αθβθ = (1, 2, 3)(2, 3) = (1, 3), and α2βθ = α2θβθ = (1, 3, 2)(2, 3) = (1, 2). It is now easily seen that θ is a bijection and (4.1) holds for all elements of D3;we say that D3 and S3 are isomorphic. Isomorphisms are bijections; on the other hand, there is much to be gained by considering maps that preserve the group operation (they satisfy (4.1)) but are not necessarily bijective, these maps are called homomorphisms. For example, suppose ∗ G1 = GL2(Q), G2 = Q and φ : G1 → G2 is defined by Aφ = det A for A ∈ G1. As det A = 0 and det AB = det A det B, for all A,B ∈ G1,themapφ is a homomor- phism, that is, it satisfies (4.1). We shall refer to this example repeatedly in the next few sections, and so we call it the standard example. In this chapter, we define homomorphisms, isomorphisms and factor groups, de- rive their basic properties—the Isomorphism and Correspondence Theorems, dis- cuss the cyclic groups, and introduce automorphisms (that is, isomorphisms of a group to itself). There are two Web Sections, 4.6 and 4.7, the first discusses a special kind of homomorphism called the transfer used in Web Section 6.5, and the second extends our work on presentations introduced in Chapter 3. Maps—Left or Right In algebraic contexts, we write maps and functions on the right, and in most cases we do not use brackets; that is, we write aφ and not φ(a) forthevalueofthemapφ at the argument a. In the western world, we read and write from left to right, and so this is a more natural notation—when applying a map φ to an argument a,wefirst choose a in the domain, then we apply φ to obtain the value aφ in the range (co-domain); see Appendix A. This notational convention may seem strange at first, but it does make many constructions clearer, especially those involving composition or permutations. Also we use lower case Greek letters for maps or functions (including permutations) throughout. The argument of a map φ is given by the Roman letter or letters immediately to its left. For example, in the expression abcφ the argument is abc, but in the expression aφbψ the argument of ψ is b (and this expression is the product of aφ and bψ). On some occasions we use brackets to aid clarity, so we might write (abc)φ for abcφ,or(aφ)(bψ) for aφbψ,but(aφb)ψ when the argument of ψ is aφb, that is when the argument is the product of aφ and b. 4.1 Homomorphisms and Isomorphisms 69 4.1 Homomorphisms and Isomorphisms The basic notions are given by Definition 4.1 Let G1 and G2 be groups, and let φ be a map from G1 to G2.The map φ is called a homomorphism from G1 to G2 if ghφ = gφhφ for all g,h ∈ G1. Note that the product gh on the left-hand side of this equation is in G1, and the product gφhφ (or (gφ)(hφ)) on the right-hand side is in G2. Definition 4.2 Let φ be a homomorphism mapping G1 to G2. (i) φ is called the trivial homomorphism if aφ = e, for all a ∈ G1; (ii) φ is called an isomorphism if it is also a bijection from G1 to G2, the groups 1 G1 and G2 aresaidtobeisomorphic, and we write G1 G2 in this case; (iii) φ is called an endomorphism if it is a homomorphism of G1 to itself; (iv) φ is called an automorphism if it is an isomorphism of G1 to itself. Isomorphisms were first introduced by Jordan in 1865 whilst he was working on his proof of the Jordan–Hölder Theorem; see Chapter 9. We use the word ‘trivial’ in (i) above to imply that the map trivialises, or destroys, all properties of the group except those associated with the neutral element. The identity map ι : G → G which is given by gι = g for all g ∈ G is an example of an automorphism of G. The reader should also refer to the note on ‘isomorphism classes’ on page 17. Examples One homomorphism was discussed on page 68 (the standard example), four more are given now; see also the examples on pages 17 and 84, and in Prob- lem 4.1. (a) If G1 = R, G2 is the group of complex numbers having absolute value 1 with the operation complex multiplication, and φ1 : G1 → G2 where xφ1 = cos x + i sin x, for x ∈ R, then φ1 is a homomorphism mapping G1 to G2. It is not an isomor- phism because xφ1 = (x + 2kπ)φ1 for each k ∈ Z. (b) If G1 = Z/6Z (the set {0, 1, 2, 3, 4, 5} with operation addition modulo 6), G2 = (Z/7Z)∗ (the set {1, 2, 3, 4, 5, 6} with operation multiplication modulo 7), and φ2 : G1 → G2 is given by a aφ2 = 3 modulo 7, for a ∈ G1, then φ2 is an isomorphism; the reader should check this. 1Some authors use the symbol =∼ in place of . 70 4 Homomorphisms ∗ (c) If G1 = C , the multiplication group of the non-zero compex numbers, and φ3 : G1 → G1 satisfies 2 zφ3 = z , then φ3 is an endomorphism. Reader, why is φ3 not an automorphism? (d) If G1 = Z and φ4 : G1 → G1 satisfies aφ4 =−a for a ∈ G1, then φ4 is an automorphism of G1. The basic properties of these maps are given by the following lemmas. Lemma 4.3 Suppose φ is a homomorphism between the groups G1 and G2 with neutral elements e1 and e2, respectively. (i) e1φ = e2. −1 −1 (ii) If g ∈ G1 then g φ = (gφ) . Note that in (ii) the inverse on the left-hand side is in G1, and the inverse on the right-hand side is in G2. Proof (i) As g = ge1 for all g ∈ G1,wehavegφ = ge1φ = gφe1φ and (i) fol- lows by cancellation. (ii) Using the group axioms and (i) we have −1 −1 e2 = e1φ = gg φ = gφg φ, the result follows as inverses are unique and two-sided (Theorem 2.5). From now on we use the symbol e for the neutral element of every group. Lemma 4.4 Suppose φ : G1 → G2, ψ : G2 → G3, and both φ and ψ are homo- morphisms. (i) φ ◦ ψ is a homomorphism mapping G1 to G3. (ii) The image of φ is a subgroup of G2. (iii) If H ≤ G1 and φ is the map φ with its domain restricted to H , then φ is a homomorphism from H into G2. (iv) If φ is an isomorphism, so is φ−1. The image of φ, see (ii), is denoted by im φ.Alsoφ in (iii) is sometimes written φ|H and described as ‘φ restricted to H ’. Proof Straightforward, see Problem 4.3. We now define the kernel, an important entity in the theory. It is a normal sub- group, the reader should review the material of these subgroups given in Chapter 2. 4.1 Homomorphisms and Isomorphisms 71 Definition 4.5 Let φ be a homomorphism mapping G1 to G2. The subset of G1 defined by {a ∈ G1 : aφ = e} is called the kernel of φ, and it is denoted by ker φ. ∗ In the standard example φ : GL2(Q) → Q where Aφ = det A (page 68), ker φ is the set of 2 × 2 matrices A with determinant 1, the neutral element of Q∗. Hence ker φ equals SL2(Q) in this example. (Note SL2(Q) GL2(Q).) The basic properties of the kernel are given by Lemma 4.6 Suppose φ : G1 → G2 is a homomorphism. (i) ker φ G1, see Definition 4.5. (ii) φ is injective if and only if ker φ = e . This lemma is useful in its own right, but it is also useful because it provides a second method (apart from Theorem 2.29) for showing that a subset of a group is a normal subgroup, that is, by showing the subset in question is the kernel of a homomorphism; see, for example, the proof of the N/C-theorem (Theorem 5.26). Proof (i) The property ker φ ≤ G1 follows from Theorem 2.13, Defini- tion 4.1, and Lemma 4.3. Now by Lemma 4.3 again, if a ∈ G and g ∈ ker φ, a−1gaφ = a−1φgφaφ = (aφ)−1eaφ = e, which shows that a−1ga ∈ ker φ. Normality follows by Theorem 2.29. (ii) Suppose first φ is injective. If ker φ = e , we can find c ∈ G1 such that c = e and cφ = e = eφ; but as φ is injective, this implies c = e which contradicts our assumption. Hence ker φ = e . Conversely, if ker φ = e and bφ = cφ,forb,c ∈ G1, then − − − e = (cφ) 1bφ = c 1bφ, and so c 1b ∈ ker φ. But ker φ = e , and so b = c. This holds for all b,c ∈ G1, and so the result follows. As an example we give a proof of Cayley’s Theorem. It was first proved in 1850, and was a development of some work undertaken by Cauchy during the previous decade; see page 42. Note that there is nothing unique about the symmetric group here, for it can be shown that every group of order n is isomorphic to a group of m × m matrices, for some m ≤ n, defined over an arbitrarily given field, see Prob- lem 4.17. Also, in Web Section 4.7 we show that every group is a factor group of a free group. 72 4 Homomorphisms Theorem 4.7 (Cayley’s Theorem) Every group is isomorphic to a subgroup of a symmetric group. Proof Let G be a group. For fixed g ∈ G, define the map σg : G → G by aσg = ag for all a ∈ G. Using cancellation (Theorem 2.6), we see immediately that σg is a bijection; that is, σg is a permutation of the underlying set of G. Also, if g,h ∈ G, aσgh = a(gh) = (ag)h = (aσg)σh = a(σg ◦ σh) for all a ∈ G. (4.2) Hence we can define a map θ : G → SG, where SG is the group of permuta- tions of the elements of the set G,by gθ = σg for all g ∈ G, and it is a homomorphism by (4.2). The result follows by Lemma 4.4(ii). This result is usually not the best possible. For example, the group D4 has order 8 and so, by Cayley’s Theorem, it is isomorphic to a subgroup of S8 (with order 40320). But, in fact, it is isomorphic to a subgroup of S4 (with order 24), see Chap- ter 8. On the other hand, for a few groups the theorem is best possible, for example, Q2 (which also has order 8) is isomorphic to no subgroup of Sn if n<8. Factor Groups Cosets were defined in Chapter 2.Hereweask: Is it possible to make the set of cosets of a subgroup H of G into a new group? The answer is yes, but only when H is a normal subgroup of G (Definition 2.28). Consider the following simple example which is typical of the general situation. Let G = Z and H = 2Z, the even integers under addition. Clearly H G, and there are just two cosets: the even integers 2Z, and the odd integers 2Z + 1. Now an even integer plus an even integer is even, an even integer plus an odd integer is odd, and (∗) an odd integer plus an odd integer is even. This suggests that we can treat the set of cosets in this example as a new two element group with the operation given by (∗), and this will characterise the terms ‘even’ and ‘odd’ when applied to the integers. Hence we make the following 4.1 Homomorphisms and Isomorphisms 73 Definition 4.8 Given K G, g,h ∈ G and using the operation in G, we define the coset product,orcoset multiplication,ofgK and hK by gKhK = ghK. There is an important point concerning this definition which can cause some misunderstanding at first. By Lemma 2.22,ifj ∈ gK then jK = gK; that is, the representative g in the coset gK is not unique. Therefore, in the definition above we must check (in Theorem 4.9 below) that it is a product of cosets considered as entities in their own right, and it does not depend on the coset representatives g and h—we say the product is “well-defined”. Theorem 4.9 Suppose K G. (i) If aK = a K and bK = b K, then abK = a b K. (ii) The set of cosets of K in G with coset multiplication given by Definition 4.8 forms a group. Proof (i) The hypotheses and Lemma 2.22 give k1,k2 ∈ K to satisfy a = ak1 and b = bk2, which shows that a b = ak1bk2. By Theorem 2.29 and as K is a normal subgroup, we can find k3 ∈ K to satisfy k1b = bk3,so a b = ak1bk2 = abk3k2 ∈ abK, and (i) follows by Lemma 2.22 again. (ii) By (i), the set of cosets is closed under (well-defined) coset multiplica- tion. Using the corresponding properties of G, coset multiplication is associa- tive, the neutral element is K (as eK = K), and the inverse of the coset aK is a−1K (as aKa−1K = aa−1K = K); the result follows. We give an example to show that normality is essential in this result. Let G = 3 2 2 D3 = c,d | c = d = e, dc = c d and let J = d .WehavecJ ={c,cd} and Jc ={c,c2d}, that is, J is not normal in G. Property (i) also fails, for if we let a = c, a = cd and b = b = c, then aJ = cJ ={c,cd}=a J (as d2 = e), and bJ = cJ = b J .ButabJ = c2J ={c2,c2d} whilst a b J = cdcJ ={d,e}=J which shows that abJ = a b J . Definition 4.10 The group of cosets given by Theorem 4.9(ii) is called the factor group, or sometimes the quotient group,ofG by K, and it is denoted by G/K. 74 4 Homomorphisms In some contexts, G/K is referred to as “G over K”. We shall show below, especially in Theorem 4.17, that this ‘fractional’ notation is a reasonable choice but it needs to be treated with care. However, we can see already by Lagrange’s Theorem (Theorem 2.27) that, if o(G) < ∞, then o(G/K) = o(G)/o(K). Example The notation Z/nZ used in Chapter 2 can now be explained. As nZ Z (the groups are Abelian) we can form the factor group Z/nZ using Definitions 4.8 and 4.10. By the First Isomorphism Theorem to be proved below, this group is iso- morphic to the group N of integers modulo n with addition modulo n as its op- eration, see page 18. The factor group Z/nZ contains n cosets, they are r + nZ, for r = 0,...,n− 1, and the coset product mirrors the standard addition modulo n in N exactly. As Z/nZ is isomorphic to N, we use this (distinctive) notation for both—a slight ‘abuse of the notation’, but as the groups are isomorphic, no problems should arise. 4.2 Isomorphism Theorems We come now to what is probably the single most important collection of results in the theory—the Isomorphism Theorems. Note that the naming and numbering of the theorems given below is not universally accepted. Although many of the ideas, theorems and proofs had been ‘known’ for some time previously, they were first sys- tematically formulated and proved in detail2 by the German mathematician Emmy Noether (1882–1935) during the 1920s; and one of their first appearances in print was in Moderne Algebra by B.L. van der Waerden. This two-volume work was first published in the 1930s and it has had a considerable impact on the development of algebra in general during the twentieth century. We begin by returning to the example concerning even and odd integers discussed on page 72. Define a map φ : Z → T1 (page 12)by aφ = 1ifa is even, and aφ =−1ifa is odd. Clearly, this is a surjective homomorphism mapping Z to T1 with kernel 2Z, and Z/2Z T1, see (∗) on page 72. This is typical of the general situation given by the following major result. 2A proof of the First Theorem appears in Burnside’s classic text written a quarter of a century before; see also the comment below Definition 4.2. 4.2 Isomorphism Theorems 75 Theorem 4.11 (First Isomorphism Theorem) If G1 and G2 are groups, and φ : G1 → G2 is a surjective homomorphism with kernel ker φ = K, then G1/K G2. Proof By Theorem 4.9 and Lemma 4.6, G1/K is a group. We define a map θ : G1/K → G2 as follows. If a ∈ G1 then aK ∈ G1/K, and we set (aK)θ = aφ. First, we need to show that θ is well-defined. Suppose aK = a K, then by Lemma 2.22 we have a a−1 ∈ K = ker φ, and so a a−1φ = e which gives a φ = aφ. This shows that aKθ = aφ = a φ = a Kθ as required. The map θ is surjective because φ is surjective. It is also injective, for if aKθ = bKθ, then by definition aφ = bφ which gives in turn e = (aφ)−1bφ = a−1bφ and a−1b ∈ ker φ = K. By Lemma 2.22 again, this shows that aK = bK,soθ is injective, and therefore it is a bijection. For the homomorphism property we have (aK)θ (bK)θ = aφbφ by definition = abφ as φ is a homomorphism = (abK)θ by definition = (aK)(bK) θ by coset product, and the theorem follows. All is not lost if the homomorphism is not surjective, for we have the corollary given below where the symbol stands for ‘is isomorphic to a subgroup of’. Corollary 4.12 If ψ : G1 → G2 is a homomorphism with kernel ker ψ = K, then G1/K G2. Proof By Theorem 4.11, G1/K im ψ, and by Lemma 4.4(ii) we have im ψ ≤ G2, the corollary follows. If in this corollary the group G1 is simple, then it is isomorphic to a subgroup of G2 provided ψ is not the trivial homomorphism. ∗ ∗ Examples Let G1 = C , G2 = R , see page 18, and let φ be the absolute value function given by ∗ zφ =|z| for z ∈ C . This is a non-surjective homomorphism with image R+.Nowkerφ is the subgroup of C∗ of those complex numbers which have absolute value 1. Corollary 4.12 gives 76 4 Homomorphisms ∗ ∗ C / ker φ R ; that is, associated with every non-zero complex number there is a unique non-zero real number, its absolute value. This value belongs to R+ a subgroup of R∗. Returning to the standard example we note that the det function is a surjective ho- ∗ momorphism mapping GLn(Q) onto Q with kernel SLn(Q), hence Theorem 4.11 gives ∗ GLn(Q)/SLn(Q) Q . (4.3) Also if we replace the field Q by a finite field F with o(F) = q, then a similar isomorphism applies and we obtain o(GLn(F )) = o(SLn(F ))(q − 1) by Lagrange’s Theorem (Theorem 2.27)asF has q − 1 non-zero elements. We single out for special mention the following homomorphism. Definition 4.13 Let K G, and let φ be the surjective homomorphism G → G/K given by gφ = gK for all g ∈ G, where K = ker φ.Themapφ is called the natural homomorphism, from G to the factor group G/K = im φ. There are three further isomorphism theorems, see also Section 9.1. Some will not be required until later but as they are all consequences of the First Theorem (Theorem 4.11) we shall present them now. The Second and Third Theorems give conditions under which factor groups can be simplified, that is, parts cancelled out, whilst the remaining result, called the Correspondence Theorem, is not a single theorem but a collection of results which relate the properties of G ‘above a normal subgroup K’ to the properties of G/K, it or one of its extensions will be used many times in the sequel. We begin with a lemma about intersections and we restate the basic facts concerning products, see Theorem 2.30. Lemma 4.14 Suppose H ≤ G and K G. (i) H ∩ K H . (ii) HK ≤ G and HK = KH. (iii) HK G if we also have H G. (iv) If J H then JK HK. Proof (i) By Theorem 2.15,wehaveH ∩ K ≤ H . For normality we argue as follows. Let h ∈ H .Ifj ∈ H then h−1jh∈ H .AlsoasK G, h−1jh∈ K if j ∈ K. Hence if j ∈ H ∩ K, then h−1jh∈ H ∩ K and the result follows by Theorem 2.29. For the remaining proofs, see Theorem 2.30 for (ii) and (iii), and Prob- lem 4.6(iv) for (iv). 4.2 Isomorphism Theorems 77 We come now to the Second Theorem, as it will be used mainly in our work on series in Chapters 9 to 11, it can be omitted on a first reading of the early chapters. Theorem 4.15 (Second Isomorphism Theorem) If H ≤ G and K G, then K HK ≤ G and H/H ∩ K HK/K. Proof As K G, by Lemma 4.14 we have HK ≤ G. Also clearly K ≤ HK, and so Problem 2.14(ii) gives K HK and we can form the factor group HK/K. We define a map θ : H → HK/K by hθ = hK for h ∈ H. As hkK = hK for all h ∈ H and k ∈ K, it follows that θ is a surjective map. It is also a homomorphism for if h1,h2 ∈ H we have using coset multiplication (Definition 4.8) h1h2θ = h1h2K = h1Kh2K = h1θh2θ. Hence the conditions for the First Isomorphism Theorem (Theorem 4.11) ap- ply, and so to prove the theorem we need to show that ker θ = H ∩ K.Now h ∈ ker θ if, and only if, hK = K. If this equation holds then h ∈ K, and so h ∈ H ∩ K. Conversely, if h ∈ H ∩ K, then h ∈ K and hK = K clearly follows. Therefore, Theorem 4.11 now gives H/ker θ = H/H ∩ K HK/K. Note that one consequence of this result is: If H ≤ G, K G and H ∩ K = e , then HK/K H . Example We give a proof of a permutation result to illustrate the use of this theorem, see page 49. We show that if G ≤ Sn and G contains an odd permutation, then exactly half of the elements of G are even and half are odd. By Theorem 3.11, An Sn, and G ≤ Sn by hypothesis, so the Second Isomorphism Theorem gives G/(G ∩ An) GAn/An = Sn/An. This last equation follows because GAn contains all even and at least one odd per- mutation, and so has an order larger than o(Sn)/2, which gives GAn = Sn by Prob- lem 2.19.Now,aso(Sn/An) = 2 (Theorem 3.11), we have o(G ∩ An) = o(G)/2; that is exactly half of the elements of G belong to An.SoifG is simple and G Correspondence Theorem The Correspondence Theorem which some authors call the ‘Third Isomorphism Theorem’ will be considered now. It is an important result with many applications, and it is a direct consequence of the First Isomorphism Theorem (Theorem 4.11). As noted above, it has a number of parts, some of which will be added later. Suppose θ is a surjective homomorphism mapping G1 to G2 with ker θ = K; that is, θ maps G1 onto G2, and K onto e . Informally, the theorem says that the properties and structure of that part of G1 which lies above K is exactly mirrored by the properties and structure of G2, and vice versa. This is illustrated in the diagram below. θ G1 −→ G2 ·· θ H −→ Hθ and G1/H G2/H θ if H G1 ·· −1 θ −1 Jθ −→ J and G2/J G1/J θ if J G2 ·· θ K −→ e There are also some upward inclusions. If H satisfies K ≤ H ≤ G1, then Hθ ≤ G2, −1 and if H G1, then Hθ G1;alsoifJ ≤ G2, then K ≤ Jθ ≤ G1, et cetera. Hence we have Theorem 4.16 (Correspondence Theorem) Suppose G1 and G2 are groups, H ≤ G1, J ≤ G2, and θ is a surjective homomorphism mapping G1 to G2 with ker θ = K. This implies G1θ G2 and Kθ = e . (i) If K ≤ H ≤ G1, then Hθ ≤ G2. (ii) If K ≤ H G1, then Hθ G2 and G1/H G2/H θ. −1 (iii) K ≤ Jθ ≤ G1. −1 −1 (iv) If J G2, then Jθ G1 and G1/J θ G2/J . Proof The proof is mainly a matter of checking that group axioms or sub- group conditions hold, we shall establish parts (i) and (iv) and leave the re- maining parts for the reader to complete (Problem 4.13). (i) By Lemma 4.3, Hθ is a non-empty subset of G2. Also, if h1,h2 ∈ −1 = −1 ∈ H , then (h1θ) h2θ h1 h2θ Hθ as H is a subgroup of G1 and θ is a homomorphism. This gives (i). −1 (iv) Suppose g ∈ G1 and k ∈ Jθ , and so gθ ∈ G2 and kθ ∈ J .AsJ G2, these properties show that (gθ)−1kθgθ ∈ J, 4.2 Isomorphism Theorems 79 but (gθ)−1kθgθ = (g−1kg)θ, hence g−1kg ∈ Jθ−1. By (iii), this gives −1 Jθ G1. For the second part, define the map ψ : G1 → G2/J by gψ = (gθ)J for g ∈ G1. Now ψ is surjective as θ is surjective, and it is a homomorphism because g1g2ψ = (g1g2θ)J = (g1θ)(g2θ)J = (g1θJ )(g2θJ)= g1ψg2ψ, for g1,g2 ∈ G1. The kernel of ψ is {g ∈ G1 : (gθ)J = J }. By Lemma 2.22, − (gθ)J = J if and only if gθ ∈ J if and only if g ∈ Jθ 1. −1 As this holds for all g ∈ G1, we obtain ker ψ = Jθ . The result follows by applying the First Isomorphism Theorem (Theorem 4.11)toψ. Example Suppose G1 = Z (and so all subgroups are normal as this group is Abelian) and G2 = Z/30Z. Secondly, suppose φ represents congruence reduc- tion modulo 30, its kernel K equals 30Z. Now if, for example, H = 10Z, then K H G1, Hθ Z/10Z and both G1/H and G2/H θ are isomorphic to the cyclic group of order 3. As an exercise the reader should take J Z/15Z and apply parts (iii) and (iv) of Theorem 4.16. The last isomorphism theorem follows easily from (iv) in the result above. It provides a further justification for the factor group notation. Theorem 4.17 (Third Isomorphism Theorem) If K G and K H G, then G/H (G/K)/(H/K). This is sometimes called the Freshman’s Theorem; see Scott (1964). Proof In (iv) of Theorem 4.16, put G2 = G/K, J = H/K, and let θ be the natural homomorphism (Definition 4.13) from G to G/K. Note that H/K G/K, the reader should check this. We also have − (H/K)θ 1 = H. (4.4) For if g ∈ (H/K)θ −1, then gθ = hK for some h ∈ H . But by definition gθ = gK, and so hK = gK, and g ∈ hK ⊆ H by Lemma 2.22. This gives (H/K)θ −1 ⊆ H and, as the reverse inclusion is given by definition, (4.4) follows. Finally, as the conditions in Theorem 4.16(iv) now apply, the result follows. Referring to the example above, if Cm denotes the cyclic group of order m,we have G1/K C30,G1/H C3 and H/K C10, and Theorem 4.17 shows that C3 C30/C10 (that is 3 = 30/10 !). 80 4 Homomorphisms 4.3 Cyclic Groups In this section, we discuss the cyclic groups first introduced in Definition 2.19.The elements of a cyclic group are the powers (positive and negative) of a single gener- ator a.IfG is infinite, then all powers are distinct and G is the free group on the set {a},butifo(G) = n and a ∈ G, then an = e, see Theorem 4.20 below. The first theorem provides the basic facts. Theorem 4.18 (i) For each positive integer n there exists a cyclic group of order n. (ii) All cyclic groups of order n are isomorphic. (iii) All infinite cyclic groups are isomorphic to the group Z. (iv) Every homomorphic image of a cyclic group is cyclic. Proof (i) We give two examples. The group of integers modulo n, Z/nZ,see page 74, is cyclic of order n. Also the group of the complex numbers e2kπi/n for 0 ≤ k at φ = bt for t ∈ Z, then φ is clearly an isomorphism between G and H . (iii) This is similar to (ii). (iv) If G = a and θ : G → H is a surjective homomorphism, then H = aθ , that is H is cyclic with generator aθ. n We write Cn for the abstract cyclic group of order n, that is, Cn a | a = e , a group with a single generator a, say, and a single relation an = e; see the intro- duction to Chapter 3.AlsoweuseZ as a notation for an infinite cyclic group, see (iii) above. Theorem 4.19 (i) The subgroups of Z are e and nZ, one for each positive inte- ger n. (ii) All non-neutral subgroups of an infinite cyclic group are isomorphic to Z. Proof Suppose H contains a non-zero integer, a,b ∈ H , and H ≤ Z. Then ma + nb ∈ H for all m, n ∈ Z. We can choose m and n so that the greatest common divisor c,say,ofa and b satisfies c = ma + nb, c | a, c | b and c>0, hence c ∈ H ; see Appendix B. (We usually use the notation (a, b) for c.) This shows that the least positive integer d,say,inH is a divisor of every element of H , and so H = dZ, an infinite cyclic group. Both parts of the result follow by Theorem 4.18(iii). Theorem 4.20 Suppose G is a cyclic group of order n. It contains a cyclic subgroup H of order m if and only if m divides n, and when this happens H is unique. 4.4 Automorphism Groups 81 This and the previous result show that all subgroups of a cyclic group are cyclic. Also the proof below is not the shortest or most elegant for this result but it is given as an illustration of the Isomorphism Theorems ‘in action’. Proof We use the First Isomorphism and Correspondence Theorems. Define amapψ : Z → Cn as follows. Let j be a generator of the group Cn, and for let aψ = j c where a ≡ c(mod n) and 0 ≤ c We shall see later that the properties of cyclic groups given in Theorems 4.19 and 4.20 are special. In general, a group of order n can have many subgroups of order m, where m | n, or none at all. If m = pr ,n= ms and p s, then the subgroup Cm is called the (unique) ‘Sylow p-subgroup’ of Cn; see Section 6.2.Alsomore generally, if m | n and (m, n/m) = 1, then Cm is called a ‘Hall’ subgroup of Cn. 4.4 Automorphism Groups Let G be a group and let Aut G denote the set of all automorphisms of G,see Definition 4.2. This set can be given a group structure using composition as the operation. The composition of two bijections is a bijection (Appendix A) and so the operation is well-defined and closed by Lemma 4.4(i). Composition is associative (Appendix A), the identity map ι on G acts as the neutral element, and the inverse of an isomorphism is also an isomorphism (Lemma 4.4(iv)). Thus Aut G forms a group under composition. Definition 4.21 For a group G, the set of automorphisms of G with the operation of composition forms a new group called the automorphism group of G which is denoted by Aut G. We give an example now and another extended one at the end of the section, we shall also discuss the automorphism groups of the finite cyclic groups. Example If G Z, then Aut G C2 because there are only two automorphisms: For all g ∈ G, the first maps g → g, and the second maps g → −g; see Prob- lem 4.19. 82 4 Homomorphisms Some automorphisms are given by conjugation (Definition 2.28). Let h ∈ G be fixed, then the map τh : G → G given by −1 gτh = h gh, for g ∈ G, is an automorphism of G, the reader should check this. An automorphism of this type is called inner, and the set of all inner automorphisms of G is denoted by Inn G. It is a subgroup of Aut G as the following lemma shows. Lemma 4.22 (i) If G is a group, then Aut G is also a group. (ii) Inn G Aut G. Proof (i) This was proved above. (ii) The identity map is an inner automorphism (put h = e in the definition), so Inn G is not empty. For g,h ∈ G,letτg and τh be the corresponding inner automorphisms, and let a ∈ G. Then −1 −1 a(τgh) = (gh) a(gh) = h (aτg)h = (aτg)τh = a(τg ◦ τh). As this holds for all a ∈ G, we see that τg ◦ τh = τgh ∈ Inn(G).Also −1 = ≤ ∈ (τg) τg−1 (reader, check). Hence Inn G Aut G. Normality. Let a,g G, θ ∈ Aut G, and aθ−1 = b, that is, bθ = a. By Lemma 4.3(ii), −1 −1 −1 a θ τgθ = aθ τg θ = (bτg)θ = g bg θ −1 −1 = g θbθgθ = (gθ) a(gθ) = aτgθ. −1 This holds for all a ∈ G, hence θ τgθ = τgθ ∈ Inn G, the result follows. Using Lemma 4.22, we can form the factor group Aut G/ Inn G;itiscalledthe outer automorphism group denoted by Out G. Following normal practice, an auto- morphism which is not inner is called outer, so an outer automorphism is an el- ement of one of the cosets of Aut G/ Inn G except Inn G itself. Note that if G is Abelian then Inn G = e , and so all automorphisms except the identity automor- phism are outer. Also Corollary 5.27 gives a formula for Inn G. It was conjectured by O. Schreier (1901–1929) that Out G is soluble (for a definition see Section 11.1) if G is simple. This has been shown to be true for finite groups but only by us- ing CFSG (which is discussed in Chapter 12). The conjecture is false for non- 3 simple groups, for example, Out((C2) ) GL3(2), a non-Abelian simple group; see page 264. For a particular group, it can be major task to find its automorphism group. Some examples are given in Chapter 8. For example, consider Aut Sn. It has been shown that, if n = 2 or 6, then Aut Sn Sn and all automorphisms are inner (Problem 8.5). This is not true for S6 which does possess outer automorphisms, and the order of Aut S6 is twice the order of S6; see Problem 4.20. An account of these results is given in Rotman (1994), pages 156 to 162. 4.4 Automorphism Groups 83 Next in this section we show how to construct the automorphism group of a cyclic group. In some cases, it is also cyclic but not always. ∗ Theorem 4.23 Aut Cn (Z/nZ) . Proof Let g be a generator of Cn. A homomorphism mapping Cn to itself will map g to some power of g, so we define, for m ∈ Z, m θm : Cn → Cn by gθm = g . r r rm Note that as θm is a homomorphism g θm = (gθm) = g for all integers r. As gn = e we only need to consider m in the range 0 ≤ m r rm 1−sn g θm = g = g = g, n rt t as g = e. This shows that g θm = g for t = 1,...,n, hence θm is surjective. u um Conversely, if θm is surjective, then g = g θm = g ,forsomeu ∈{1,...,n}. This implies that g1−um = e and so 1 − um is a multiple of n. This can only happen if (m, n) = 1. Therefore, θm is an automorphism if and only ∗ if (m, n) = 1. Hence the map θm → m for m ∈ (Z/nZ) gives the required isomorphism. The groups (Z/nZ)∗ are Abelian, in Chapter 7 we shall show that they can be expressed as ‘direct products’ of cyclic groups. Also using the theory of primitive roots, a topic from number theory discussed briefly in Appendix B,wehave The group (Z/nZ)∗ is cyclic if and only if n = 2, 4,ps ,or2ps where p is an odd prime and s is a positive integer, and their orders are as follows: o((Z/2Z)∗) = 1, o((Z/4Z)∗) = 2, and o((Z/psZ)∗) = o((Z/2psZ)∗) = ps−1(p − 1). So in particular, the automorphism group of Cps where p is an odd prime and s is a positive integer is itself cyclic and has order ps−1(p − 1); a result due to Gauss. Further automorphism results are given in Problem 4.18, for elementary Abelian groups, and in Problems 6.18 and 7.10. Overleaf we end this section with an extended example which illustrates some of the methods used to construct automorphism groups, it also provides some more examples of isomorphisms. 84 4 Homomorphisms Example Show that Aut D4 D4 (also see Problem 4.19). 4 2 3 Let the dihedral group D4 be given by a,b | a = b = e, bab = a .Notefirst that an automorphism maps an element of order k to an element of order k (this follows from Problem 4.5(i) and Lemma 4.4(iv)). Also D4 has exactly two elements of order 4, that is a and a3, and so automorphisms either map a → a and a3 → a3, or map a → a3 and a3 → a. In either case, a2 → a2, and so a2 is fixed by all automorphisms. Secondly, note that b,ab,a2b and a3b all have order two and so an automorphism could map b to b,ortoab,ortoa2b,ortoa3b. This suggests defining φ and ψ by φ : a → a and b → ab, ψ : a → a3 and b → b. These give + akφ = ak,akbφ = ak 1b, and akblψ = a3kbl, where k is to be read modulo 4, and l modulo 2. It is now an easy exercise to check that both φ and ψ are automorphisms, the reader should do this. The remaining automorphisms can be generated using φ and ψ. Clearly, ψ2 = ι, the identity map (automorphism). Also bφ2 = (bφ)φ = abφ = aφbφ = a2b, and similarly bφ3 = a3b and bφ4 = b, and so φ4 = ψ2 = ι. For the remaining condition, we have aψφψ = (a3φ)ψ = a3ψ = (aψ)3 = a9 = a = aφ3, bψφψ = (bφ)ψ = abψ = aψbψ = a3b = bφ3. This shows that ψφψ = φ3. No other automorphisms are possible, see the com- ments in the paragraph above, and so the result follows. Unlike both S3 and S4 (Problems 4.19(i) and 8.5), D4 has both inner and outer automorphisms, see Corollary 5.27.Asa−1aa = a and a−1ba = a2b, we see that the automorphism φ2 corresponds to conjugation by a, and as bab = a3 and b3 = b, the automorphism ψ corresponds to conjugation by b.Butφ has no such corre- spondence, and so forms an outer automorphism. By Lemma 4.22(ii), the set of inner automorphisms forms a normal subgroup of Aut D4 which is isomorphic to 2 2 2 2 2 the 4-group T2 (page 19) in this case because (φ ) = ψ = ι and φ ψ = ψφ .The four outer automorphisms are φ,φ3,φψ and φ3ψ. 4.5 Problems Problem 4.1 Show that the following maps are homomorphisms. (i) The maps (a), (b), (c), and (d) given on pages 69 and 70. (ii) The trivial map (Definition 4.2). (iii) The sgn map from Sn to C2 (Definition 3.7 on page 47). 4.5 Problems 85 ∗ (iv) The determinant map from GL2(F ) to F where F is a field (page 68). (v) The projection map from R2 to R given by (x, y) → x. Problem 4.2 Show that the following pairs of groups are isomorphic. (i) GL2(2) and S3, see Problem 2.20. (ii) Let F be a field. The first group is F ∗, the multiplicative group of F , and the second group has underlying set F1 = F \{1} and the operation ∗ where a ∗ b = a + b − ab for a,b ∈ F1. First, you will need to show that (F1, ∗) is a group. (iii) Q+ and the additive group of all polynomials in the variable x with integer coefficients. (Hint. Use Unique Factorisation (Theorem B.6).) Problem 4.3 (i) to (iv) Give proofs of the four parts of Lemma 4.4. (v) Suppose G is a group and X is a set. Given a bijection θ : G → X, construct an operation on X so that θ becomes an isomorphism of G to the group formed by X with this operation. Show also that this operation is unique. Problem 4.4 An exercise working with cosets. Let G = SL2(3). (i) Find Z(G) and show that it has order 2. (ii) Write out a list of representatives of the cosets of Z(G) in G. (iii) Show that if E is a coset of Z(G) in G and E = Z(G), then either E2 = Z(G) or E3 = Z(G), where Ek is defined using the coset product. Count the number of solutions in each case. (iv) Use (iii) and Problem 3.10 to show that G/Z(G) A4, that is, SL2(3) can be treated as an extension (Definition 9.9)ofC2 by A4. A similar argument can be used to show that the general linear group GL2(3) is an extension of C2 by S4. Problem 4.5 (i) Suppose φ : G → G satisfies gφ = g−1 for all g ∈ G. Show that φ is a homomorphism if and only if G is Abelian. (ii) Let G be a finite group and let θ : G → G be an automorphism. Further suppose (a) if g ∈ G and gθ = g, then g = e, and (b) θ 2 is the identity map ι on G. Use these to show that gθ = g−1 for all g ∈ G, and so deduce that G is Abelian. (Hint. First show that {a−1 · aθ : a ∈ G}=G.) Problem 4.6 (Abelian Factor Groups and the Derived Subgroup) Before tackling this problem the reader should revisit Problem 2.16 which gives the basic properties of the derived subgroup. (i) Show that every factor group of an Abelian group is Abelian. (ii) Prove that if K G, then G/K is Abelian if and only if G ≤ K, 86 4 Homomorphisms where G denotes the derived (or commutator) subgroup of G.Thisisanim- portant fact we use many times. = (iii) Use (ii) to show that Sn An. (iv) Suppose H1,H2 ≤ G, H2 H1 and J G. Show that JH2 JH1, and JH1/J H2 is Abelian if H1/H2 is Abelian. Problem 4.7 Suppose K G and o(G/K) = n<∞. (i) Show that if g ∈ G, then gn ∈ K. (ii) Suppose (m, n) = 1, g ∈ G and gm ∈ K, prove that g ∈ K. Problem 4.8 (Perfect Groups) A group G is called perfect if it equals its derived subgroup, that is, if G = G . Show that (i) An equivalent definition is: No non-neutral factor group is Abelian. (ii) If H ≤ G and H is perfect, then H ≤ G . Problem 4.9 Let θ : G → H be a homomorphism. Prove the following: (i) If a ∈ G, then (an)θ = (aθ)n, for all n ∈ Z. (ii) If K G, K ⊆ ker θ and, for a ∈ G, θ : G/K → H is defined by (aK)θ = aθ, then θ is a homomorphism. You need to show that θ is well-defined. (iii) If j1,j2 ∈ G, then [j1,j2]θ =[j1θ,j2θ]. Problem 4.10 Suppose G is a finite group with the property (ab)n = anbn, for all a,b ∈ G where n is some fixed integer larger than 1. (i) Let n n n Gn = a ∈ G : a = e and G = c : c ∈ G . n Using a suitably chosen homomorphism show that both Gn and G are normal n subgroups of G, and deduce o(G ) =[G : Gn]. (ii) Show that (a) if n = 2 then G is Abelian, and (b) if n = 3 and 3 o(G), then G is again Abelian. More is known, see Alperin (1969). Problem 4.11 Let G be a finite group with a normal subgroup K satisfying o(K),[G : K] = 1. Using the Isomorphism Theorems show that K is the unique subgroup of G with order o(K) by considering what happens to another such subgroup K1 in G/K;see 4.5 Problems 87 Theorem 6.10.Ifπ denotes the set of prime factors of o(K), then K is called the π-radical of G, see Section 10.2, it is usually denoted by Oπ (G). Problem 4.12 Suppose K1,...,Kn are normal subgroups of G.LetL = G/K1 ×···×G/Kn, the direct product of G/K1,...,G/Kn; see Section 7.1.(We can treat L as the group of ordered n-tuples {g1K1,...,gnKn} for gi ∈ G with component-wise multiplication.) Define a map θ : G → L by gθ = (gK1,...,gKn) for g ∈ G. n Show that θ is a homomorphism, and deduce G/ i=1 Ki is isomorphic to a sub- group of L. Problem 4.13 (i) Give proofs of the second and third parts of the Correspondence Theorem 4.16. (ii) Suppose K G. Prove the following extension of the Correspondence The- orem: G/K is simple if and only if K is a maximal normal subgroup of G; that is, K is a proper normal subgroup of G, and no other normal subgroup J of G exists which satisfies K Problem 4.14 (Coset Enlargement) (i) Suppose J and K are normal subgroups of G, and J ≤ K.Letξ : G/J → G/K be defined by (aJ )ξ = aK for a ∈ G. Show that ξ is a well-defined surjective homomorphism with kernel K/J.Themapξ is called the enlargement of cosets map. (ii) Use (i) to reprove the Third Isomorphism Theorem (Theorem 4.17). Problem 4.15 Let G be a finite Abelian group. (i) If p | o(G), show how to find an element g ∈ G of order p.(Hint.Write o(G) = pn and use induction on n.) A second proof of this result is given in Theorem 6.2. (ii) If o(G) = mn, (m, n) = 1, H,J ≤ G, o(H) = m, o(J) = n and both H and J are cyclic, show that G is also cyclic. (iii) Give an example to show that the statement in (ii) is false if G is not Abelian. Problem 4.16 (Properties of the Centre) Derive the following properties of the centre Z(G) of a group G. Note that by Problem 2.14(ii) a subgroup of Z(G) is normal in G. Suppose K G throughout. (i) Construct an example to illustrate the following fact: There exists at least one group G with proper subgroups H and J which have the properties: e (iii) If o(K) = 2 then K ≤ Z(G). (iv) If θ : G → J is a surjective homomorphism and H ≤ Z(G), then Hθ ≤ Z(J). (v) Show that if H is an Abelian subgroup of G, then HZ(G) is also an Abelian subgroup of G. (vi) Z(K) G; does it follow that Z(K) ≤ Z(G) (see Problem 4.22)? (vii) If K ≤ J ≤ G, then [J,G]≤K if and only if J/K ≤ Z(G/K). (viii) Finally, show that Z(G) G ≤ Z . Z(G) ∩ K K Problem 4.17 Suppose F is a field and G is a finite group. Show that G is iso- morphic to a subgroup of the general linear group GLn(F ),forsomen not larger than o(G), using the following method. Let U denote the collection of all expres- = ∈ sions of the form z g∈G mgg where mg F . Apply component-wise addition and scalar multiplication to show that U forms a vector space over F . Secondly, for h ∈ G define a map θh : U → U by zθh = mggh. g∈G Show that θh defines an invertible linear map on U, and the collection of these maps forms a group isomorphic to GLn(F ) where n is the dimension of the vec- tor space U. This result can of treated as a ‘matrix version’ of Cayley’s Theorem (Theorem 4.7), note that there is no restriction on the choice of field F . Problem 4.18 (Elementary Abelian Groups) An Abelian group all of whose non- neutral elements have order p, for some fixed prime p, is called an elementary Abelian group (or sometimes an elementary Abelian p-group; see Chapter 6). (i) Let F denote the field Z/pZ (page 18), and let G be an elementary Abelian p-group. On G define an ‘addition’ ⊕ by x ⊕ y = xy,forx,y ∈ G, and a ‘scalar multiplication’ expressed using concatenation by cx = xc,forc ∈ F and x ∈ G. Prove that G with these new operations forms a vector space over F . We denote it by G. (ii) Show that the subgroups of G correspond to the subspaces of G. (iii) If G is finite, then G will have a finite basis and finite dimension m,say.(This can be proved using some basic linear algebra, or it follows from Theorem 6.3.) Show that the automorphisms of G correspond to the linear maps on G; and so deduce that Aut G GLm(p). In Problem 7.14, we shall show that G is a direct product of m copies of Z/pZ,this will provide another proof of the result given in (i). 4.5 Problems 89 Problem 4.19 (i) Find the automorphism groups of the following groups: (a) Z, (b) the 4-group T2,(c)S3,(d)C4, and (e) Cpn where p is an odd prime and n is a positive integer. (ii) Is Aut(D8) D8? (iii) Suppose o(G) < ∞. Show that Aut(G) = e if and only if o(G) ≤ 2. (Hint. Show that the map defined by a → a−1 is an automorphism of an Abelian group.) Problem 4.20 (i) Let the map ψ : S6 → S6 satisfy: (1, 2)ψ = (1, 5)(2, 3)(4, 6), (1, 3)ψ = (1, 4)(2, 6)(3, 5), (1, 4)ψ = (1, 3)(2, 4)(5, 6), (1, 5)ψ = (1, 2)(3, 6)(4, 5), (1, 6)ψ = (1, 6)(2, 5)(3, 4). Beginning with Problem 3.1(i), this can be extended to an automorphism of S6; you are not asked to prove this but you could consider what is needed. Using this 2 fact show that ψ equals the identity map on S6, and so provides an example of an outer automorphism of this group. For more details, see Rotman (1994), pages 156 to 167. It can be shown that Aut Sn Sn provided n = 2 or 6. When n = 2 use (iii) in Problem 4.18. But it is a fact that S6 is the only non-Abelian symmetric group which has an outer automorphism; see page 82. (ii) Use (i) and Problem 3.21 to show that S6 has 12 subgroups isomorphic to S5, note that six of them possess no 2-cycles and act transitively on six points. One member of this second set of six subgroups can be constructed as follows: Let H be the group generated by (1,i + 1)ψ for i = 1, 2, 3, 4, then choose four (of ten) products of three 2-cycles in H which satisfy the conditions of Problem 3.21 with n = 4. Problem 4.21 The outer automorphism group for the group A5 is isomorphic to C2. Find an outer automorphism for A5, and consider what would be needed to establish the previous statement. (Hint. Use Theorem 3.6.) Problem 4.22 (Project—Characteristic Subgroups) In this project, you are asked to develop the notion of characteristic subgroup which is similar to, but stronger than, normality. A subgroup H of a group G is said to be characteristic in G if Hφ ≤ H for all φ ∈ Aut G; this is denoted H char G. (Note that Hφ = {hφ : h ∈ H }.) Prove the following statements. (i) It is sufficient to require Hφ= H . (ii) Normality is equivalent to: Hν ≤ H for all inner automorphisms ν : G → G. 90 4 Homomorphisms (iii) Unlike normality, the characteristic relation is transitive. (iv) If J char K and K G, then J G, but does this proposition also follow if J K and K char G? (v) A subgroup of a cyclic group is characteristic. (vi) Z(G) char G; we say “the centre of a group is a characteristic subgroup of G”. (vii) The derived subgroup G of G is a characteristic subgroup of G. (viii) Give an example of a normal subgroup which is not characteristic. Chapter 5 Action and the Orbit–Stabiliser Theorem There is only a small intersection (mainly involving examples) between the material in this chapter and the next, with that in Chapter 7. Hence Chapter 7, which contains work on direct products and Abelian groups, can be read first. In the last chapter, we introduced homomorphisms, they are maps that transfer properties from one group to another, and they satisfy the homomorphism equa- tion (4.1). Here, given a set X we introduce new collections of maps that transform X to itself and which are governed by a group G; that is, for each g ∈ G we de- fine a map \g : X → X, and map composition ‘corresponds’ to the group operation. The map \g is a permutation of X and it is called an action of G on X.Inmany cases, X is closely related to G, but not always. It is also possible to define an ac- tion using homomorphisms; see Theorem 5.12. In one sense, this important notion has been part of the theory since its inception, but only in particular instances. If you look for the word ‘action’ in Scott’s group theory text published in 1964—the standard introduction to the theory at that time—you will not find it. But you will find many entities now associated with actions, for example, ‘centraliser’ and ‘nor- maliser’. About this time, and due in part to the work of Wielandt (1964), it was realised that a number of constructions have a similar basis, and emphasising their similarity would give new insights into the theory. Also at around this time, elegant proofs of some major theorems based on new and easily defined actions appeared; for instance, McKay’s proof of Cauchy’s Theorem (Theorem 6.2) or the main proof of the First Sylow Theorem (Theorem 6.7) both given in the next chapter. Nowadays actions form an important part to any introduction to group theory. We begin this chapter by defining actions and two major associated entities: or- bits and stabilisers. We then prove the OrbitÐStabiliser Theorem which leads to a number of important applications. Three major examples follow that introduce centralisers and normalisers, vital entities especially for the finite theory. In Web Section 5.4, we extend our work on permutations begun in Chapter 3, discuss transitive and primitive permutations, and prove Iwasawa’s simplicity lemma, this work has applications in Chapter 12. H.E. Rose, A Course on Finite Groups, 91 Universitext, DOI 10.1007/978-1-84882-889-6_5, © Springer-Verlag London Limited 2009 92 5 Action and the OrbitÐStabiliser Theorem 5.1 Actions We begin by considering an example. Let G be the group (Z/7Z)∗, and let X = {1, 2, 3, 4, 5, 6}. In this particular example, the set X and the group G have the same elements, and so we have underlined these elements when they are being treated as members of X. Consider the (right) multiplication of an element x of X by an element g of G, that is, x · g. Later we shall write this as x\g.Wehave x · e = x, that is, the (right) multiplication of elements of X by e does not alter X. Also, by associativity we have x · (gh) = (x · g) · h, where g,h ∈ G, that is, applying gh to x is the same as first applying g to x, and then applying h to the result. We call this procedure an action of G on X,see Definition 5.1 below. Further, as 4 ∈ G we have 1 · 4 = 4, 2 · 4 = 1, 3 · 4 = 5, 4 · 4 = 2, 5 · 4 = 6 and 6 · 4 = 3, that is, the set X has been permuted by this procedure. There is nothing special about ‘4’, the calculation works for all elements of G; the reader should try one. This procedure gives a map from the group G to the set SX of all permutations of X, where X is the underlying set of G; see Theorem 5.2 below. The term G-set is sometimes used for the set X (that is, when G is acting on X), we will not use this notation because in some cases X and G are unrelated. With the example above in mind we begin by stating the basic Definition 5.1 Given a non-empty set X and a group G,wesayG acts on X if, for each g ∈ G, there exists a map \g : X → X, and these maps satisfy (i) x\e = x, (5.1) (ii) x\(gh) = (x\g)\h, for all x ∈ X and g,h ∈ G. We call the map \g an instance of the action of the group G on the set X. Notes (a) By (5.1), the group operation ‘corresponds’ to composition of actions, that is, the map \gh is defined to equal \g ◦\h. (b) We usually follow the convention that groups and elements of groups are denoted by letters at the beginning of the alphabet, and sets and elements of sets are denoted by letters at the end of the alphabet. (c) More formally, we can rewrite Definition 5.1 as follows: The function \ that maps the set of pairs X × G to X, and which satisfies the two parts of (5.1), is called an action of G on X, see Theorem 5.12. (d) The action defined above is a ‘right action’; we could also define a left action but as we write functions on the right, we shall only consider the former. 5.1 Actions 93 (e) Many authors use either xg, or sometimes xg,forx\g. Both concatenation and the exponential notation are used widely in many contexts, so for the sake of clarity, it seems preferable to use a new symbol, those readers used to the old con- catenation notation can simply ignore the backslash. Personally, the author finds the exponential notation particularly confusing. Also we use the concatenation notation xg for the particular action called the natural action (Example (a) below), and so we need a distinct notation for the general case. In a recently published book by Isaacs (2008), the author uses x • g for our x\g. We shall give some examples below, but first we prove the following basic result. Theorem 5.2 Using the notation set out above, the map \g : X → X is a permuta- tion (bijection) of the set X. Proof Suppose x,y ∈ X and x\g = y\g. Then by (5.1)wehave x = x\e = x\ gg−1 = (x\g)\g−1 = (y\g)\g−1 = y\ gg−1 = y, that is, the map \g is injective. Secondly, suppose z ∈ X then, for g ∈ G, z = z\e = z\ g−1g = z\g−1 \g, that is, z\g−1 is a preimage of z under the map \g. Hence this map is also surjective, and so it is a permutation of X. Examples We give four here, three more extended examples will be discussed in the next section; see also the proofs of Cauchy’s and the First Sylow Theorems given in Chapter 6. (a) Let G be a group and let X be the underlying set of G. The group G acts on X by right multiplication if we define, for g ∈ G and x ∈ X, x\g = xg, see the example given at the beginning of this section. This clearly satisfies the conditions (5.1), and the corresponding action is called the natural action on G. (b) Let V be a vector space defined over a field F . The multiplicative group F ∗ of F acts on V by scalar multiplication,forifa,b ∈ F ∗ and v ∈ V , the standard vector space axioms give v\1 = v1 = v and v\(ab) = v(ab) = (va)b = (v\a)\b. (c) Let G =e and X be an arbitrary set, then G acts on X if we define x\e to equal x for all x ∈ X. (d) Let G = Sn and let X ={1,...,n}, then if we define x\σ = xσ for σ ∈ G and x ∈ X, we obtain a new action called the permutation action; see Theorem 5.2. 94 5 Action and the OrbitÐStabiliser Theorem There are two important entities associated with an action which govern its basic properties. They are orbits and stabilisers, we introduce orbits first. Let the group G act on the set X. Define a relation ∼ on X by: If x,y ∈ X, then x ∼ y if and only if x\g = y for some g ∈ G. To put this informally, x is related to y if we can ‘get from x to y’ by using an element of G. Lemma 5.3 The relation ∼ defined above is an equivalence relation. Proof We have x ∼ x as x\e = x. Secondly, suppose x ∼ y, that is, x\g = y for some g ∈ G. Then y\g−1 = (x\g)\g−1 = x\ gg−1 = x\e = x, and so y ∼ x. Finally, suppose we also have y ∼ z with y\h = z for some h ∈ G. Then x\(gh) = (x\g)\h = y\h = z, that is, x ∼ z. Definition 5.4 (i) An equivalence class of the equivalence relation ∼ given in Lemma 5.3 above is called an orbit of the action of G on X. The orbit contain- ing the element x ∈ X is called the orbit of x, and it is denoted by OG{x},orO{x} when it is clear which group G is being used. (ii) An action of G on X is called transitive if there is only one orbit, that is, X itself, otherwise it is called intransitive. By Lemma 5.3, X is a disjoint union of its orbits (Appendix A). The orbit of x ∈ X, O{x}, is the subset of X of those elements that we can ‘get to’ starting with x and applying elements of G (that is, by applying the maps associated with the elements of G), so an action is transitive if we can ‘get from’ every member of X to every other member of X by applying elements of G. For instance, the action in Example (d) opposite is transitive because Sn contains all 2-cycles—if y,z ∈ X, then the 2-cycle (y, z) belongs to Sn and this 2-cycle maps y to z.But if in this example we change the group to (1, 2) C2, then the action would not be transitive if n>2 because, for instance, no element of this group maps 1 to 3. Also, if we replace X by X1 ={1,...,n,n+ 1}, the action of Sn on X1 would again be intransitive because no permutation in Sn maps 1 to n + 1. An extension of transitivity is as follows. An action of G on X is called k-transitive if for all pairs of k-element subsets Y and Z of X, there exists an element g ∈ G such that \g is a bijection between Y and Z. For example, the action given in Example (d) is k-transitive if k ≤ n. See Web Section 5.4 and Section 12.4. We have defined both orbits and cycles (Definition 3.3), the notion of an orbit in a general group is related to the notion of a cycle in a symmetric group. If n>1 and 5.1 Actions 95 σ ∈ Sn then, by Theorem 3.4, σ can be expressed as a (disjoint) product of cycles σ = (a1,...,aj )(b1,...,bk) ··· where j + k +···=n.LetH =σ , the cyclic subgroup of Sn generated by σ , then H acts on X ={1,...,n} by setting x\h = xh for h ∈ H and x ∈ X.Hereh has the t t form σ for some t ∈ Z, and so the orbit of a1,say,is{a1σ : t = 0, 1, 2,...} which is exactly the cycle (in σ ) containing a1. Hence in this particular example, orbits and cycles coincide. Our second new entity is the stabiliser which we introduce by Definition 5.5 GivenagroupG actingonasetX (Definition 5.1), and x ∈ X,the subset of G, g ∈ G : x\g = x , is called the stabiliser of x in G; it is denoted by stabG(x). The stabiliser of x is the subset (subgroup, see Lemma 5.6)ofG of those elements whose associated maps ‘do not move’ x. For instance, if we let x = 1 in Example (d) on page 93, then the stabiliser of 1 is the set of all permutations in Sn which leave the element 1 fixed. This is clearly a subgroup of Sn isomorphic to Sn−1, and is an instance of Lemma 5.6 Using the notation set out in Definition 5.5, for x ∈ X, stabG(x) ≤ G. Proof Clearly, e ∈ stabG(x) as x\e = x by (5.1). If g,h ∈ stabG(x), then x\g = x = x\h and, by (5.1)again,wehave x\ gh−1 = (x\g)\h−1 = (x\h)\h−1 = x. −1 This shows that gh ∈ stabG(x), now use Theorem 2.13. We come now to the OrbitÐStabiliser Theorem, the main result in this chapter. It has a number of applications and, considering its importance, it is remarkably easy to prove. Theorem 5.7 (OrbitÐStabiliser Theorem) If G acts on a set X, x ∈ X, and OG{x} is the orbit of x, then o OG{x} = G : stabG(x) . 96 5 Action and the OrbitÐStabiliser Theorem Proof Note first stabG(x) ≤ G by Lemma 5.6. We define a map γ from OG{x} to the set of right cosets of stabG(x) in G, and the theorem follows by showing that this map is a bijection. The map γ is given by (x\g)γ = stabG(x) g, where g ∈ G, and so x\g ∈ OG{x}. First, we show that this map is well- defined. Suppose x\g = x\h, then as in the proof above we have x\ gh−1 = (x\g)\h−1 = (x\h)\h−1 = x\e = x −1 by (5.1), that is, gh ∈ stabG(x). By Lemma 2.22, this gives (stabG(x))g = (stabG(x))h, as required. Second, note that γ is clearly surjective, for if g ∈ G, then one preimage of (stabG(x))g is x\g. Last, we show that it is injective. Suppose stabG(x) g = stabG(x) h, −1 −1 then, as above, this shows gh ∈ stabG(x), and so x\(gh ) = x. Hence x\h = x\gh−1 \h = x\ gh−1h = x\g, that is, γ is injective. The theorem now follows. Referring back to Example (d) on page 93,letx = n. The orbit of n is {1,...,n} O{ } = (as the action is transitive), and so o( n ) n. We noted above that stabSn (n) Sn−1. Hence we have by the OrbitÐStabiliser Theorem, and as o(Sn) = n!, O{ } = =[ : ]= : o n n Sn Sn−1 Sn stabSn (1) . This is a easy example but it does provide an illustration of the theorem ‘at work’. As a second application of this theorem we prove the following useful result. Another proof was given in Problem 2.27. Note that if H and J are both non-normal subgroups of G, then HJ is not a subgroup of G, see the example below. But if either H or J is normal, then HJ is a subgroup (Theorem 2.30) and the result follows using the Second Isomorphism and Lagrange’s Theorems. Theorem 5.8 If G is a finite group and H,J ≤ G, then o(H J )o(H ∩ J)= o(H)o(J). Proof We define an action. Let X ={Hg : g ∈ G}, the set of right cosets of H in G. The subgroup J acts on X by right multiplication if we set Hg\j = Hgj for j ∈ J. This is an action since Hg\e = Hge = Hg and Hg\j1j2 = Hgj1j2 = (Hgj1)\j2 = (Hg\j1)\j2. The orbit O{H} of H is {H \j : j ∈ J }, and this 5.1 Actions 97 equals HJ, and so o(HJ) = o(H) × o(O{H}). (Note that HJ is a disjoint (by Lemma 2.23) union of right cosets of H , and the orbit of H under this action is the union of those cosets of H we can ‘get to’ starting with H it- self and applying elements j in J .) Further, the stabiliser of H ,stabJ (H ), equals {j ∈ J : Hj = H}, and so stabJ (H ) = H ∩ J (as Hj = H if and only if j ∈ H ). Hence, using the equation for o(HJ) above, the OrbitÐStabiliser Theorem gives o(HJ) o(J) = o O{H} = J : stab (H ) = , o(H) J o(H ∩ J) using the equation for stabJ (H ) and Lagrange’s Theorem (Theorem 2.27)for the last identity, the result follows. 3 2 2 Example Suppose G = D3 a,b | a = b = e, a b = ba, H =b≤G and J =ab≤G. Then o(H) = o(J) = 2, H ∩ J =e, and so o(HJ) = 4, by Theo- rem 5.8. Clearly, HJ is not a subgroup of G (as 4 6, also neither H nor J is nor- mal), but it is a union of two cosets. For as ab = ba2,wehaveHab= Hba2 = Ha2 and so HJ = H ∪ Ha2. Note that we also have HJ = J ∪ bJ . We have shown (Theorem 5.2) that an action of a group G on a set X is a col- lection of permutations of X; that is, the action provides a map from G to SX.We shall develop this further. Definition 5.9 Let the group G act on the set X.Themapν : G → SX given by gν =\g, for all g ∈ G, is called the permutation representation of G for this action. Lemma 5.10 The map ν given by Definition 5.9 is a homomorphism. Proof For all g,h ∈ G and x ∈ X,wehaveby(5.1), x\(gh) = (x\g)\h = x(\g ◦\h) by the definition of composition (◦) of functions. The lemma follows as this holds for all x ∈ X. This leads to the following useful Theorem 5.11 Let G act on X with permutation representation ν as defined above, then ker ν = stabG(x). x∈X 98 5 Action and the OrbitÐStabiliser Theorem Proof This is an immediate consequence of the definitions. As ν is a homo- morphism (Lemma 5.10), its kernel is the set of those g ∈ G for which \g is \ = ∈ the identity permutation in SX, that is, x g x for all x X. But stabG(x) is ∈ \ = the set of those g G for which x g x, hence x∈X stabG(x) is the set of those g ∈ G for which x\g = x for all x ∈ X; that is, the kernel of ν. ∈{ } Referring again to Example (d) on page 93,ifk 1,...,n , then stabSn (k) is the set of all permutations which fix k. Hence the intersection of these stabilisers for k = 1,...,n,ise, and so the kernel of the permutation representation in this case is the neutral subgroup. This reflects the fact that Sn has very few normal sub- groups. The converse of the last result is also valid as we show now. It can be used as an alternative definition of the action of a group G on a set X. Theorem 5.12 Suppose σ : G → SX is a homomorphism of G to the group of all permutations on the set X. The map defined by \g = gσ, for all g ∈ G, is an action of G on X, and the permutation representation of this action is identical to σ . Proof For x ∈ X,wehavex(eσ) = x (as eσ is the identity permutation ι on X) and, by composition of maps and as σ is a homomorphism, x(gσ) (hσ ) = x(gσ ◦ hσ ) = x (gh)σ . Hence, if we define x\g = x(gσ), for all g ∈ G and x ∈ X, these equations show that \g is an action of G on X.Letν be the permutation representation of this action, that is, for g ∈ G and x ∈ X, x\g = x(gν). Combining these facts gives x(gν) = x(gσ) for all x ∈ X, hence gν = gσ for all g ∈ G, which shows that ν = σ . Restricted Actions Here we ask: What is the relationship of stabH (x) to stabG(x) when H ≤ G?To answer this question we first need to consider the subset of those elements which are fixed by an action. Definition 5.13 Let G act on the set X.Weset fix(G, X) = x ∈ X : x\g = x for all g ∈ G ; it is called the fixed set of X under the action of G. 5.2 Three Important Examples 99 Note that fix(G, X) is a subset of X, and so it is not a group; for example, it is empty when the action is transitive, also we have the equivalent definitions fix(G, X) = x ∈ X : O{x}={x} = x ∈ X : stabG(x) = G . (5.2) If G and X are given by the first example in this chapter (page 92), then fix(G, X) =∅, but if we change X to X ={1,...,7}, then fix(G, X) ={7}. Let G act on a set X and let H ≤ G,wesayH acts on X by restriction of the action of G on X if we ignore those maps \g in the action of G on X where g ∈ H , and only consider those maps \h where h ∈ H . This is clearly an action because H is a (sub)group. For example, if G = Z, H = 2Z, X is the underlying set of G (the integers) and the action of G on X is the natural one given by x\g = xg, then the orbit of x under the action of G is the set of all integer multiples of x, whilst the orbit of x under the restricted action (by H ) is the set of all even integer multiples of x. We have, for x ∈ X and H ≤ G, stabH (x) = stabG(x) ∩ H, (5.3) by definition of the restricted action. We also have Lemma 5.14 If H ≤ G, G acts on X, and H acts on X by restriction of the action of G, then x ∈ fix(H, X) if and only if H ≤ stabG(x). Proof We have x ∈ fix(H, X) if and only if stabH (x) = H by (5.2) if and only if stabG(x) ∩ H = H by (5.3) if and only if H ≤ stabG(x), by Problem 2.5. For an example, see Problem 5.3(i). One consequence of this problem is: If G is a p-group (Section 6.1) and p o(X), then there exists x ∈ X whichismovedbyno g ∈ G, that is, fix(G, X) is not empty. 5.2 Three Important Examples In this section we discuss three action examples, the first involves cosets, and the second and third use conjugation of elements and subgroups, respectively. All three introduce major new concepts and theorems which will be used widely in the fol- lowing chapters, and as noted above all three have a long history in the theory. 100 5 Action and the OrbitÐStabiliser Theorem Coset Action For the first example, choose H , a subgroup of G, and let X be the set of right cosets of H in G.Giveng ∈ G and Hx ∈ X, we define (H x)\g = Hxg. (5.4) This is an action because (H x)\e = Hxe= Hx and (H x)\g \h = (H xg)\h = Hxgh= (H x)\(gh). Further, it is a transitive action. For if Hx,Hy ∈ X, then (H x)\x−1y = (H x)x−1 \y = Hy, and so there is only one orbit, that is, X itself. Also −1 stabG(H x) = x Hx because stabG(H x) = g ∈ G : (H x)\g = Hx = g ∈ G : Hxg= Hx = g ∈ G : xgx−1 ∈ H = x−1Hx, by Problem 2.23. Hence, using the OrbitÐStabiliser Theorem (Theorem 5.7), we obtain −1 G : x Hx = G : stabG(H x) = o(X) =[G : H ] (5.5) because there is only one orbit which in this case is the set of right cosets of H in G, this gives another proof of Problem 2.23(ii). If νH is the permutation representation of this action, then by Theorem 5.11, −1 ker νH = x Hx, x∈G see the comment below Definition 2.21. The entity on the right-hand side of this equation is called the core of H in G, core(H ) which was defined in Problem 2.24; it is the largest normal subgroup of G contained in H .If[G : H ]=n<∞, then SX Sn, and νH gives a homomorphism from G into Sn. Hence we have Theorem 5.15 (i) If H Proof (i) This follows immediately from Theorem 5.11 and Corollary 4.12. (ii) Both of these properties follow from (i) and Problem 2.15(i) with J = core(H ). 5.2 Three Important Examples 101 This is a useful result, especially when n is small, for it shows that if a group G has only a few normal subgroups, then its total number of subgroups is also restricted. Example Subgroups of A5.IfG is simple and H We give an application of Theorem 5.15 here, more will follow later. Theorem 5.16 If G is an infinite group, H ≤ G, and [G : H ] < ∞, then G contains a normal subgroup K which satisfies K ≤ H and G/K is finite. = −1 Proof Take K g∈G g Hg in Theorem 5.15 and use Problem 2.23.The factor group G/K is finite because this theorem gives an injective homomor- phism into a finite symmetric group. This shows that if an infinite group has a subgroup of finite index (and so is infinite), then it also has a normal infinite subgroup of finite index—an important fact concerning infinite groups. This also shows that an infinite simple group has no subgroups of finite index. Centralisers and Class Equations Our second extended example involves conjugation, and introduces a number of new concepts and constructions. Let G be a group and let X be the underlying set of G.Ifg,x ∈ G, then g−1xg is called the conjugate of x by g (Definition 2.28). The group G acts on its underlying set G by conjugation if we define − x\g = g 1xg, (5.6) for all g,x ∈ G. The operation defined by (5.6) is an action; for clearly x\e = e−1xe = x, and we have x\(gh) = (gh)−1xgh = h−1 g−1xg h = h−1(x\g)h = (x\g)\h. 102 5 Action and the OrbitÐStabiliser Theorem The action (5.6) is called the conjugacy action on G, three important entities are associated with it as follows. Definition 5.17 Using the conjugacy action defined above, the orbit of x under this action is called the conjugacy class of x and it is denoted by CG{x}, that is, −1 CG{x}= g xg : g ∈ G . This is, of course, the set of conjugates of x in G. When it is clear which group is involved, we write C{x} for CG{x}. Definition 5.18 The stabiliser of x in G under the conjugacy action (5.6) is called the centraliser of x in G, it is denoted by CG(x); that is, −1 CG(x) = stabG(x) = g ∈ G : g xg = x . Note this is equivalent to CG(x) ={g ∈ G : xg = gx}; therefore, the centraliser of x in G is the subgroup (Lemma 5.6) of those elements g ∈ G which commute with x. Applying the OrbitÐStabiliser Theorem (Theorem 5.7) we have directly Theorem 5.19 Using the conjugacy action (5.6) defined above, if x ∈ G, then o CG{x} = G : CG(x) . By Lagrange’s Theorem (Theorem 2.27), this shows that the order of a conjugacy class of a finite group G divides the order of G. It also shows that the set of elements that commute with a fixed element a, say, forms a subgroup. Both of these facts have important ramifications; for the first, see Lemma 5.21 below. The third entity associated with the conjugacy action is the centre, see Defini- tion 2.32.Ifτ denotes the permutation representation of the conjugacy action de- fined above, then the kernel, ker τ , is just the centre of the group Z(G); that is, Z(G) = ker τ = CG(x). x∈G Note also that, using Definitions 2.32 and 5.17,wehave x ∈ Z(G) if and only if CG{x}={x}. (5.7) Example We construct the conjugacy classes, centralisers and centre of the dihedral group D3.LetD3 be given by (Section 3.4) a,b | a3 = b2 = e,ba = a2b . 5.2 Three Important Examples 103 We have −1 2 C{a}= g ag : g ∈ D3 = a,a ,oC{a} = 2, 2 CD (a) = g ∈ D3 : ga = ag = e,a,a =a,oCD (a) = 3, 3 3 C{b}= b,ab,a2b ,oC{b} = 3, ={ }= = CD3 (b) e,b b ,oCD3 (b) 2. The reader should check these statements and complete the remaining cases, they provide applications of the OrbitÐStabiliser Theorem. Note finally that Z(D3) =e ∩ = as CD3 (a) CD3 (b) e . Putting these ideas together we can introduce the Class Equations by Theorem 5.20 (Class Equations) Suppose G is a finite group, and CG{y1},..., CG{yk} is a complete list of the conjugacy classes of G whose orders are larger than 1. k ˙ ˙ (i)G= Z(G) ∪ CG{yi} (disjoint unions). i=1 k k (ii)o(G)= o Z(G) + o CG{yi} = o Z(G) + G : CG(yi) . i=1 i=1 Proof (i) This follows immediately from Lemma 5.3 and (5.7) using Defini- tion 5.17—G is a disjoint union of its conjugacy classes. (ii) The first equation is given by (i) as the unions are disjoint, and the second follows from Theorem 5.19. The equations in (i) and (ii) above are called the Class Equations for G.Also the positive integer o(Z(G)) + k, that is, the number of conjugacy classes of G including the singleton classes counted in o(Z(G)), is called the class number of G and it is denoted by h(G). See Appendix C for some examples. Next we give some applications of the Class Equations, more will follow later. Lemma 5.21 If p is prime and o(G) = pn, then Z(G) = e. This lemma has important implications for p-group theory; see Section 6.1. Proof If G is Abelian there is nothing to prove, and if not, then at least one yi exists with o(CG{yi})>1. Referring to the notation given in Theorem 5.20, we have by Theorems 5.19 and 2.27 p G : CG(yi) , for i = 1,...,k, as these indices are larger than 1 by definition of yi . Hence by the second Class Equation we have p | o(Z(G)) because p | o(G) by hypoth- esis. Now o(Z(G)) > 0(ase ∈ Z(G)), and so this shows that o(Z(G)) ≥ p and therefore o(Z(G)) cannot equal 1; the result follows. 104 5 Action and the OrbitÐStabiliser Theorem Theorem 5.22 (i) If o(G) = pn, then G is not simple, provided n>1. (ii) If p is prime and o(G) = p2, then G is Abelian. If o(G) = p, then G is simple and cyclic (and so Abelian) by Theorem 2.34, and (ii) deals with the case o(G) = p2. But there exist non-Abelian groups with order pn for n>2; see Section 6.1 and Problem 6.16. Proof (i) This follows by Lemmas 2.31 and 5.21 as e In Chapter 7, we show that if o(G) = p2, then G is cyclic or a ‘product’ of two cyclic groups of order p, that is, elementary Abelian, see Problem 4.18. Centralisers played a vital role in the solution of CFSG, see Chapter 12. Brauer and Fowler (1955) showed that for a given finite group G there can only be finitely many simple groups H which contain an involution (element of order 2) a and have the property: CH (a) G. In many cases, G can be taken to be quite small and only a few simple groups H are involved, several simple groups can be characterised using this result; see the discussion on page 265. We can extend the notion of centraliser of subsets of a group as follows. Definition 5.23 The centraliser CG(X) of a subset X of a group G is given by CG(X) = g ∈ G : ag = ga for all a ∈ X . The basic properties are as follows; proofs are left as exercises for the reader, see Problem 5.8. Note that as the size of X increases, the size of CG(X) decreases. (i) If X ⊆ G then CG(X) ≤ G. = (ii) CG(G) Z(G). ≤ = (iii) If H G then CG(H ) h∈H CG(h). (iv) If J ≤ H ≤ G and H ≤ CG(J ), then J ≤ Z(H). If we refer back to the example on page 102 we see that, using (iii) above, = = { } = CD3 ( a ) a , CD3 ( b ) b and CD3 ( a,b ) e as the reader can easily check. See also Problem 5.26. 5.2 Three Important Examples 105 Normaliser Our last extended action example also uses conjugation but now applied to sub- groups. This will introduce the normaliser—one of the most important entities in group theory, and one with a long history in the theory. If H ≤ G and g ∈ G, then g−1Hg = g−1hg : h ∈ H , is a subgroup of G which is called a conjugate subgroup of H in G (Problem 2.23). Using this notion we can define an action of G on the set of all subgroups H of G by − H\g = g 1Hg for g ∈ G. (5.8) This is an action because H\e = e−1He = H and, for g,h ∈ G, H \gh = (gh)−1Hgh = h−1(g−1Hg)h = (H \g)\h. The orbit of H under this action is the set of subgroups of G which are conjugate to H in G (Problem 2.23 again), and the stabiliser is given by Definition 5.24 For H ≤ G, the stabiliser of H in G under the action (5.8) defined above, that is, −1 stabG(H ) = g ∈ G : g Hg = H , is called the normaliser of H in G, and it is denoted by NG(H ). The normaliser NG(H ) clearly contains H , and it is the largest subgroup of G in which H is normal; the reader should check this, see Problem 5.13. Note that NG(e) = G = NG(G), so in particular it is not true in general that if H Lemma 5.25 Suppose H ≤ G. (i) H NG(H ) ≤ G. (ii) NG(H ) = G if and only if H G. (iii) The number of conjugates of H in G equals [G : NG(H )]. −1 Proof (i) Clearly H ≤ NG(H ),ash Hh= H if h ∈ H ;alsoNG(H ) ≤ G by definition and Lemma 5.6. Theorem 2.29 gives normality. −1 (ii) If NG(H ) = G use (i), and if H G, then g Hg = H for all g ∈ G, that is NG(H ) = G. (iii) This follows immediately from the OrbitÐStabiliser Theorem (Theo- rem 5.7). 106 5 Action and the OrbitÐStabiliser Theorem Example Let G = S4 and H =(1, 2, 3) C3.Wehave NG(H ) =(1, 2, 3), (1, 2) S3, for (1, 2)(1, 2, 3)(1, 2) = (1, 3, 2), and so (1, 2) ∈ NG(H ), et cetera. Hence H< NG(H ) < G. Sometimes H equals NG(H ) (in this case, we say H is self- normalising), for example, when H is a maximal non-normal subgroup of G.In other cases, H The final pair of results in this section are easily proved and have a number of useful applications; see Section 4.4 for the basic properties of automorphisms. Theorem 5.26 (N/C-Theorem) Suppose H ≤ G. (i) CG(H ) NG(H ). (ii) NG(H )/CG(H ) is isomorphic to a subgroup of Aut H . The factor group NG(H )/CG(H ) is called the automiser of H in G. −1 Proof For g ∈ G,letφg be the (inner) automorphism given by aφg = g ag for a ∈ G, and define θ : NG(H ) → Aut H by jθ = φj |H for j ∈ NG(H ); (5.9) that is, jθ is φj with its domain restricted to the subgroup H . Note that jθ ∈ −1 Aut H because j ∈ NG(H ),asj hj ∈ H for h ∈ H , and φ|H is defined by conjugation. Also θ is a homomorphism (as the reader can check). Now a ∈ ker θ if and only if φa|H is the identity map on H, if and only if h−1ah = a for all h ∈ H, if and only if a ∈ CG(H ), using Definition 5.23. Hence ker θ = CG(H ), (i) follows by Lemma 4.6, and (ii) follows by Corollary 4.12. Example For this example, the reader will need to refer to Section 8.2 where the group SL2(3) is discussed; we show here that this group has a factor group iso- A G = SL ( ) H = 22 , 12 Q morphic to 4; see Problem 3.10.Let 2 3 and 21 22 2. We have H G, and so NG(H ) = G, CG(H ) = Z(G) C2, and Aut H S4 (Problem 6.18). In this case, the N/C-theorem gives G/Z(G) Aut H S4.But o(G/Z(G)) = 12, and we have G/Z(G) A4 because the only subgroup of S4 of order 12 is A4 (Problem 3.3(vii)). 5.3 Problems 107 The last result in this chapter provides an essential first step in the construction of the automorphism group of a group. Corollary 5.27 G/Z(G) Inn G. Proof Put H = G in Theorem 5.26.WehaveNG(G) = G (as the normaliser of a subgroup always contains that subgroup), CG(G) = Z(G) (see (ii) on page 104), and θ ∈ Inn G using (5.9) with H = G. Note also that the map θ given in (5.9) is surjective in this case. For example, this corollary shows that Inn S4 S4 as this group is centreless. In fact, we have Aut S4 S4, but this is harder to prove, see Problem 8.5. Some further applications are given in Chapter 8, and in the example at the end of Section 4.4. 5.3 Problems Problem 5.1 (i) Suppose G ≤ S4 and G acts naturally on the set {1, 2, 3, 4}, see (d) on page 94. Construct the orbits and stabilisers of this action when (i) G =(1, 2, 3), (ii) G =(1, 2, 3, 4), (iii) G =(1, 2)(3, 4), (1, 3)(2, 4),(iv)G = (1, 2), (3, 4), and (v) G = A4. (vi) Find the orbits and stabilisers of the action given in Example (b) on page 93 when dim V = n and n>1. (vii) Let Q[x1,...,xn] denote the set of all polynomials with rational coefficients in the variables x1,...,xn.Forf(x1,...,xn) ∈ Q[x1,...,xn] and σ ∈ Sn, define f(x1,...,xn)\σ = f(x1σ ,...,xnσ ). Show that this defines an action of Sn on the set Q[x1,...,xn], and describe the orbits and stabilisers. Hence prove that the order of the set of n-variable polynomials of the form f(x1,...,xn)\σ , where f is fixed, is a divisor of n!. Problem 5.2 Let G be a group with subgroups H and J , and suppose we have the identity ([G : H], [G : J ]) = 1. Show that (i) G = HJ, and (ii) [G : H ∩ J ]= [G : H ][G : J ]. (Hint. Use Theorem 5.8.) Problem 5.3 Suppose o(G) = pn where p is a prime, and so G is a p-group, see Section 6.1. (i) If G acts on a finite set X using Definition 5.13 show that o fix(G, X) ≡ o(X) (mod p). (ii) Prove that if e=J and J G, then J ∩ Z(G) = e—a useful result, note that it provides a generalisation of Lemma 5.21. 108 5 Action and the OrbitÐStabiliser Theorem Problem 5.4 Suppose G acts on a set X. −1 (i) If x,y ∈ X and y = x\g for some g ∈ G, show that stabG(y) = g stabG(x)g, and so deduce the result: o(stabG(x)) = o(stabG(y)). (ii) For g ∈ G, let fix(g, X) denote the subset of X of those x which are fixed by g (so x\g = x). Show that if m is the number of orbits of this action, then m = 1/o(G) o fix(g, X) . g∈G (iii) Using (ii), show that if 1 Problem 5.5 (i) Use Theorem 5.19 to show that o(G) = 2 when G has just two conjugacy classes. (ii) Show that if G is a finite non-Abelian group, 1 < [G : H ] < 5 and H ≤ G, then G is not simple. (iii) Let G be a finite group and let h(G) denote its class number, see page 103. Show that the total number of ordered pairs (a, b) which satisfy ab = ba, where a,b ∈ G,ish(G) · o(G). Problem 5.6 If p0 is the smallest prime dividing o(G), K ≤ G and [G : K]=p0, prove that K G. (Hint. Use Theorem 5.15.) Problem 5.7 (i) If C is a conjugacy class of a group G, show that the set C−1 = {a−1 : a ∈ C} is another conjugacy class of G, and vice versa. (ii) Construct the conjugacy classes, and their inverses as given by (i), for the groups D4,A4 and SL2(3). Problem 5.8 (Properties of the Centraliser) (i) to (iv) Prove the centraliser proper- ties listed on page 104. (v) Let H ≤ G and g ∈ G. Show that CH (g) = CG(g) ∩ H , and −1 −1 g CG(H )g = CG gHg . (vi) Prove CG(H ) ≤ NG(H ) without using Theorem 5.26. (vii) Let H ≤ G. Show that CG(H ) =e if, and only if, Z(J) =e for all J satisfying H ≤ J ≤ G. Problem 5.9 (i) Find the centralisers of the elements of D6 and S4, and of the subgroups of S4. (ii) Find the normalisers of the subgroups of S4; see Section 8.1 and Lemma 5.25(iii). (iii) Write out the Class Equations (Theorem 5.20)forA5 explicitly; your answer should include a description of all centralisers involved. Problem 5.10 Let τ = (1,...,m)be an m-cycle in Sn where n ≥ m>1. Show that × = CSn (τ) τ Sn−m, where we assume that S0 e . The direct product notation × is defined in Section 7.1. 5.3 Problems 109 Problem 5.11 (i) Suppose G is a non-Abelian group. Show that if a ∈ G\Z(G), then aZ(G) is an Abelian subgroup of G which properly contains Z(G). (ii) Suppose H,J ≤ G where H is Abelian. The subgroup H is called maximal Abelian if J is not Abelian whenever H Problem 5.12 Let A and B be subsets of the group G. Show that (i) if A ⊆ B then CG(B) ≤ CG(A), (ii) A ⊆ CG(CG(A)), (iii) CG(CG(CG(A))) = CG(A). Problem 5.13 (Properties of the Normaliser) Let H ≤ G. Prove that (i) NG(H ) is the largest subgroup of G in which H is normal. −1 −1 (ii) NG(g Hg)= g NG(H )g, for all g ∈ G. (iii) If H ≤ J ≤ G, then NJ (H ) = NG(H ) ∩ J . (iv) If K ≤ G, then [H,K]≤H if, and only if, K ≤ NG(H ). (v) Let p beaprime.IfJ ≤ G, o(J) = pr for some r>0 and p |[G : J ],soJ is a non-Sylow p-subgroup of G (Chapter 6). Prove that J Problem 5.14 Let D denote the subgroup of diagonal matrices in G = GL2(Q). Find NG(D). Do the same calculation for GL3(Q). Problem 5.15 (i) Suppose H Problem 5.16 Throughout this problem K G and we say that K is central if K ≤ Z(G).UsetheN/C-theorem (Theorem 5.26) to prove the following proposi- tions. (i) If G is perfect (Problem 4.8) and K is cyclic, then K is central. (ii) If p0 is the smallest prime dividing o(G) and o(K) = p0, then K is cen- tral. (iii) If G is infinite and K is finite, then G/CG(K) is finite. Deduce that if the only finite factor group of G has order 1, and o(K) < ∞, then again K is central. Problem 5.17 (i) Suppose K H ≤ G and J = CG(K). Show that (a) H ≤ NG(K), and (b) if J is self-normalising (that is, J = NG(J )), then K ≤ Z(H). (ii) Let H ≤ G. Prove that CG(H ) = NG(H ) if and only if H ≤ Z(NG(H )). These properties are used in the statement of Burnside’s Normal Complement The- orem (Theorem 6.17). 110 5 Action and the OrbitÐStabiliser Theorem Problem 5.18 In this problem, you are asked to show that a group G of order 15 is Abelian using the Class Equations; see also Problem 7.21. The method is as follows: Assume the contrary and use Problem 4.16(ii) to show that Z(G) =e, then use the Class Equations (Theorem 5.20) to show that G has exactly one conjugacy class of order 5 consisting of elements of order 3, and then obtain a contradiction. Problem 5.19 Suppose G = GL2(3) and so o(G) = 48, see Theorem 3.15 and Problem 6.23. (i) Find the centre Z(G) and show that o(Z(G)) = 2. (ii) Let H denote the set of upper triangular matrices in G (that is, H = UT2(3) ab = = which has elements of the form 0 c where a 0 c and we work modulo 3). Show that Z(G) ≤ H ≤ G, and o(H) = 12. (iii) Prove that core(H ) = Z(G) (Problem 2.24). (iv) Use Theorem 5.15 to prove that G/Z(G) S4, see Problem 4.4. Problem 5.20 Suppose CG(a) G. Use Problem 2.25 to show that a belongs to a normal Abelian subgroup of G. Is the converse false? Problem 5.21 Suppose o(G) is finite, and a,b ∈ G. (i) Show that the number of elements g in G satisfying g−1ag = b equals o(CG(a)). (ii) Deduce o(CG(a)) ≥ o(G/G ) where G denotes the derived subgroup of G. (iii) Now suppose K G, [G : K]=p (p a prime), and c ∈ K with the property: CK (c) < CG(c). Show that if b is conjugate to c in G, then b is also conjugate to c in K. Problem 5.22 Let r = h(G) be the class number of the finite group G,see page 103. Show that (i) If p0 is the smallest prime dividing o(G) and rp0 >o(G), then Z(G) = e. (Hint. Use the Class Equations.) (ii) If G is not Abelian, then r>o(Z(G))+ 1. (Hint. Use Problem 4.16(ii).) (iii) If o(G) = p3 and G is not Abelian, then G = Z(G), o(Z(G)) = p and r = p2 + p − 1. (Hint. Apply Problem 4.6 and Lemma 2.31, and show that no conjugacy class has order p2 using Problem 5.21.) Problem 5.23 (i) Show that elements of the same conjugacy class have conjugate centralisers. (ii) If n1,...,nr is a list of the orders of the centralisers of elements of distinct −1 +···+ −1 = conjugacy classes of G, prove that n1 nr 1. (iii) Deduce there are only finitely many groups with class number r using the fact that there are only finitely many ways of writing a positive integer as a sum of reciprocals, see Problem B.4. (iv) Find all groups with class number 3. 5.3 Problems 111 Problem 5.24 Suppose G is finite and H ≤ G. Show that − o g 1Hg ≤ 1 + o(G) −[G : H ]. g∈G Use this result to show that if H ≤ G and H contains at least one element of each conjugacy class of G, then H = G. Problem 5.25 Use the following method ((a) to (d)) to show that if K An, n>4, and K contains a 3-cycle, then K = An. With Theorem 3.14 this provides a new proof of the simplicity of An when n>4. Throughout suppose σ is a 3-cycle in K. (a) Show that CSn (σ ) > CAn (σ ). = ∩ (b) Secondly, show that CAn (σ ) CSn (σ ) An. [ : ]= (c) Using Theorem 5.8 and (b), deduce CSn (σ ) CAn (σ ) 2. C { }=C { } (d) Lastly, show that Sn σ An σ , and use Theorem 3.6. Problem 5.26 (Project—Centralisers and Normalisers of Groups of Order 24) Let G1 = S4 and G2 = SL2(3), see Chapters 3 and 8. First, calculate the centralisers of each of the elements of these groups. Second, calculate the centralisers and nor- malisers of each of the subgroups. Also check that the properties (i) to (iv) given on page 104 apply. Two major theorems in the theory are connected to the relationship between subgroup centralisers and normalisers: the N/C-theorem (Theorem 5.26) and Burnside’s Normal Complement Theorem (Theorem 6.17). Check that these results apply to the groups G1 and G2, and their centralisers and normalisers. Chapter 6 p-Groups and Sylow Theory There are important connections between number theory and finite group theory, re- sults in one theory have vital applications in the other, and both theories benefit from this interaction. Lagrange’s Theorem shows that a major invariant of a finite group is its order and the prime factorisation of the order. We shall develop this aspect of the theory here; the number-theoretic results we use are discussed in Appendix B. Elements of order two play a central role, they are called involutions. One reason is that an involution is its own inverse; but this is not the only special property. For example, all groups with exponent 2 (a2 = e for all a in the group) are Abelian (Corollary 2.20), and in Problem 2.28 we showed that a non-Abelian simple group is generated by its involutions. But perhaps the most remarkable fact concerning the number two in the theory is the theorem of Feit and Thompson (1963) which states that no non-cyclic group of odd order can be simple. In fact, more is true—a group of odd order must be ‘soluble’, see Chapter 11. First in this chapter, we consider groups whose orders are powers of a particular prime number p—the p-groups. These groups have a number of useful properties, for example, they have normal subgroups for all orders dividing the group order. Also the number of groups of order pn increases exponentially with n, some details are given on page 118. We have noted previously that the converse of Lagrange’s Theorem is not true in general (page 101). But it is true for prime powers, that is, for subgroups which are p-groups. This is the beginning of the Sylow theory which we develop in the second section of the chapter; it is central to finite group theory. We shall give a number of applications. In Section 6.3, we show that if o(G) has up to three not necessarily distinct prime factors, then G is not simple and it can be determined completely; we also prove a remarkable result due to Frattini—the Frattini argument—and in- troduce ‘nilpotent groups’, which have many properties in common with p-groups including being ‘reverse Lagrange’. We give some more applications in Web Sec- tion 6.5, these will include a proof of Burnside’s Normal Complement Theorem (Theorem 6.17) and a description of those groups all of whose Sylow subgroups are cyclic, see page 130. Throughout this chapter p denotes a prime number. H.E. Rose, A Course on Finite Groups, 113 Universitext, DOI 10.1007/978-1-84882-889-6_6, © Springer-Verlag London Limited 2009 114 6 p-Groups and Sylow Theory 6.1 Finite p-Groups In this section, we develop the basic properties of finite p-groups, and begin with Definition 6.1 Let p be a fixed prime. A group G is called a p-group if all of its elements have orders which are powers of p. The neutral group e is a p-group for all primes p as p0 = 1. We shall give some more examples at the end of this section. In the finite case, Definition 6.1 can be replaced by G is a finite p-group if and only if o(G) is a power of p. This follows from Cauchy’s Theorem (Theorem 6.2) for which we give three proofs. The first is due to J. McKay and uses a simple action argument, the second uses the Class Equations (Theorem 5.20) but it has the disadvantage that the Abelian case must be treated separately. The result also follows directly from the First Sylow Theorem (Theorem 6.7). Cauchy proved this result for permutation groups in 1845 as part of a series of papers on the properties of permutations; see Section 3.1.Inall probability, the first proof for general groups was given by Jordan in the 1870s. Theorem 6.2 (Cauchy’s Theorem) If G is a finite group, p is a prime, and p | o(G), then G contains at least one element of order p. In fact, G contains at least p − 1 elements of order p. In the case p = 2, there may be a unique element of order 2, for instance, in the quaternion (dicyclic) group Q2 (page 119) or the special linear group SL2(3) (page 172). Note that Cauchy’s Theorem applies to all finite groups. Proof We begin by introducing a new action. Let X = (g1,...,gp) : gi ∈ G, for i = 1,...,p, and g1 ···gp = e , that is, X is the set of all ordered p-tuples of elements of G whose product is the neutral element. We prove the theorem by applying an action of the group Z/pZ to X, and counting orbits. Note first − o(X) = o(G)p 1. (6.1) = −1 ··· −1 ∈ For arbitrary g1,...,gp−1 in G,ifwetakegp gp−1 g1 , then gp G, the p-tuple (g1,...,gp) belongs to X, and gp is unique by Theorem 2.5. We define an action of Z/pZ (the integers with addition modulo p)onX as follows. If (g1,...,gp) ∈ X and a ∈ Z/pZ, then (g1,...,gp)\a = (ga+1,...,gp,g1,...,ga). 6.1 Finite p-Groups 115 The action of a on an element of X cyclically permutes its entries by a places modulo p.ThesetX is closed under this action because, if (g1,...,gp) ∈ X (and so g1 ···gp = e), then by associativity −1 ga+1 ···gpg1 ···ga = (g1 ···ga) (g1 ···ga)(ga+1 ···gp)(g1 ···ga) = e. Also it is easily seen that the action axioms (5.1) are satisfied. By the OrbitÐStabiliser Theorem (Theorem 5.7), an orbit of this action has order 1 or p (the divisors of o(Z/pZ)). Suppose there are r orbits of order 1, and s orbits of order p. An element of an orbit of order 1 has the form (g,...,g) for some g ∈ G with gp = e (as all cyclic permutations give the same p-tuple). Now r>0 because there is at least one orbit of this type, namely the orbit of the p-tuple (e,...,e). If this was the only orbit of order 1, then r = 1, and by (6.1), − 1 + sp = o(X) = o(G)p 1, as X is the disjoint union of its orbits. But this is impossible because p | o(G) by hypothesis, and so p divides the right-hand side of the equation above whilst it clearly does not divide the left-hand side. Therefore, r>1 (in fact, r is a positive multiple of p), and so there must exist at least one g ∈ G satisfy- ing g = e and o(g) = p. We give a second proof of this theorem using the Class Equation. As noted above the Abelian case must be established first, see Problem 4.15. Second proof—Non-Abelian case We use induction on o(G). Choose a ∈ G such that a/∈ Z(G), it exists because G is not Abelian. This implies that o(C G{a})>1, and so by Theorem 5.19 [G : CG(a)] > 1, that is, CG(a) is a proper subgroup of G.Ifp | o(CG(a)), then the theorem follows by induction because o(CG(a)) < o(G) and so by the inductive hypothesis CG(a) has an element of order p. Hence we may suppose p o(CG(a)) for all a ∈ G\Z(G). But as p | o(G) by hypothesis, this implies p G : CG(a) , for all a ∈ G\Z(G) by Lagrange’s Theorem (Theorem 2.27). Applying this to the second Class Equation (Theorem 5.20)givesp | o(Z(G)).ButZ(G) is Abelian, and so by Problem 4.15, Z(G) contains an element of order p, and hence so does G. We can now show that our two definitions of a finite p-group are equivalent. Theorem 6.3 If G is finite and p is prime, then G is a p-group if and only if o(G) = pr for some non-negative integer r. 116 6 p-Groups and Sylow Theory Proof There is nothing to prove if r = 0. Suppose G is a p-group (Defini- tion 6.1)oforderqt n where q is a prime which does not divide n, t>0, and q = p. By Theorem 6.2, G contains an element of order q, which contra- dicts the hypothesis. Therefore, q = p and n = 1. The converse follows from Lagrange’s Theorem and Definition 2.19. Next we show that the property of being a p-group is preserved by taking both subgroups and factor groups, and vice versa. Very few properties are preserved in this way. Abelianness is not one, but it is true for finiteness and, as we shall show later, it is also true for solubility (Chapter 11). Theorem 6.4 (i) Subgroups and factor groups of p-groups are p-groups. (ii) If K G, and K and G/K are both p-groups, then G is also a p-group. Proof (i) The first part follows from the definition. For the second part, suppose g ∈ G and o(g) = pr , then using coset multiplication we have r r K = eK = gp K = (gK)p . Hence the order of gK is a divisor of pr , and so the order of every element of G/K is a power of p. r (ii) If g ∈ G, then (gK)p = K for some integer r by the second hy- r pothesis, hence gp ∈ K.ButK is a p-group (the first hypothesis), and so r o(gp ) = ps for some non-negative integer s, so the order of g is a divisor of pr+s . The result follows as this holds for all g ∈ G. The p-groups have many important properties, for example, they are reverse La- grange. In fact, the following stronger result follows from Lemma 5.21. Theorem 6.5 If G is a group with order pr , then G is a finite p-group (by Theo- rem 6.3) and it has subgroups G0,...,Gr satisfying (a) G0 = e , Gr = G; (b) for i = 1,...,r, Gi G and Gi−1 Gi ; i (c) for i = 0,...,r, o(Gi) = p . Proof The proof is by induction on r.Ifr = 1 there is nothing to prove, and if r = 2 the result follows from Lemma 5.21 and Theorem 5.22, for then G is Abelian. Hence we may assume that the result holds for all p-groups of order less than pr with r>1. By Lemma 5.21 again, Z(G) = e , and so Z(G) contains an element y which satisfies o(y) = pt for some positive t, and if we t−1 put z = yp , then z ∈ Z(G) and o(z) = p. Set G1 = z , we have G1 G because G1 ≤ Z(G) G, see Problem 2.14(ii). r−1 Let H = G/G1, by Lagrange’s Theorem (Theorem 2.27), o(H) = p . The inductive hypothesis provides a sequence of subgroups H0,...,Hr−1 which satisfies (a), (b) and (c) with r − 1forr, H for G, and Hi−1 for Gi . 6.1 Finite p-Groups 117 Let θ be the natural homomorphism G → H G/G1, then the Correspon- −1 dence Theorem (Theorem 4.16(iii) and (iv)) gives Gi = Hi−1θ which are subgroups of G containing G1,fori = 1,...,r − 1. Applying this theorem again for i = 0,...,r − 1, we have Gi+1 G(as Hi H) and Gi Gi+1 (as Hi−1 Hi). Parts (a) and (b) follow, and (c) follows by Lagrange’s Theorem. The groups Gi in Theorem 6.5 are in many cases not unique. If ri denotes the i number of subgroups of G with order p , both normal and non-normal, then ri ≡ 1 (mod p). We shall prove this fact for ‘Sylow’ subgroups in the next section, and the general result is given in Rotman (1994). There is an extension to Theorem 6.5 which applies when K is a maximal sub- group of G. In this case, K has prime index and is normal in G (cf. Problem 2.19(i)). We derive these properties now. Theorem 6.6 Suppose G is a finite p-group, and K is a maximal subgroup. (i) K G. (ii) [G : K]=p. Proof (i) By induction on r where o(G) = pr , the result clearly holds when r = 1. By Lemma 4.14(ii), K ≤ KZ(G) ≤ G (the second inequality holds as Z(G) G), and so by the maximality of K we have KZ(G) = G or KZ(G) = K, both are possible. If KZ(G) = G, there exists g ∈ Z(G)\K,butg ∈ NG(K) (because g ∈ Z(G)); therefore, K is a proper subgroup of NG(K) which im- plies, by the maximality of K, that NG(K) = G. In turn, this implies that K G by Lemma 5.25(ii). The second possibility is KZ(G) = K, then Z(G) K by Problem 2.14 and Lemma 2.31.AsK is a maximal subgroup of G,wealsohaveK/Z(G) is a maximal subgroup of G/Z(G) by Problem 4.13(ii). By Lemma 5.21 and as G is a p-group, we have o(Z(G)) > 1, so o G/Z(G) < o(G). Using the inductive hypothesis this gives K/Z(G) G/Z(G), and the Corre- spondence Theorem shows that K G. Hence in both cases K G. (ii) This follows easily from (i) and Theorem 6.5, see Problem 6.1. Theorems 10.6 and 10.7 on pages 213 and 214 give a second proof of (i). 118 6 p-Groups and Sylow Theory A large number of p-groups exist with order pr if r is large, many of which differ only slightly from one another. If v(n) denotes the number of (isomorphism classes of) groups of order n, then the following data has been established for powers of 2 (see Besche et al. 2001): v(4) = 2,v(8) = 5,v(16) = 14,v(32) = 51, v(64) = 267,v(128) = 2328,v(256) = 56092, v(512) = 10494213 and v(1024) = 49487365422; the second of these equations is established below and in Chapter 7. No exact for- mula is known for v(n) in general, but the following estimate has been given by Higman and Sims: 3 v(pn) = p2n /27 + O n8/3 . On the other hand, if n has a large number of prime factors and is square-free, then v(n) can be quite small. For all integers m, there are infinitely many integers n, with m prime factors, that satisfy v(n) = 1; this is a corollary of Dirichlet’s Theorem on Primes in Arithmetic Progressions; see, for instance, Rose (1999). As examples we have v(15) = 1, v(105) = v(3 · 5 · 7) = 1, and v(5865) = v(3 · 5 · 17 · 23) = 1; see the table in Appendix D. Non-Abelian Groups of Order 8 The smallest non-Abelian p-groups have order 8 (Problem 2.20), we consider these now. Groups of order p3, p>2, can be treated similarly (Problem 6.5), and Abelian p-groups will be discussed in Chapter 7.LetG be a non-Abelian group of order 8. If G contains an element of order 8, then G is cyclic, also if every non-neutral element has order 2, then G is Abelian (Corollary 2.20), hence we may assume that G contains an element a, say, of order 4 and, by Problem 2.19(i), a G (this also follows from Theorem 6.6). Choose b ∈ G\ a , then the elements of G are e,a,a2,a3,b,ab,a2b, and a3b. The reader should check that no two of these elements are equal—for example, if a = ab then by cancellation b = e.Nowasb2 ∈ G, it follows that b2 equals one of the eight elements of G listed above. If b2 = ar b, then b = ar but by defini- tion b/∈ a , and if b2 = a or a3, then o(b) = 8, and G is cyclic. Hence as we are considering the non-Abelian case, b2 = e or a2; both are possible. Also, as a G we have b−1ab = as ∈ a , for s = 0, 1, 2, or 3. If s = 0 then a = e,ifs = 1 then G is Abelian, and if s = 2 then e = a4 = (b−1ab)2 = b−1a2b, which implies that a2 = e. Hence there is only one possibility: s = 3, that is, b−1ab = a3. This gives ba3 = ab, and so bat = a4−t b for 6.2 Sylow Theory 119 t = 1, 2, 3. These calculations show that there are at most two (isomorphism classes of) non-Abelian groups of order 8: 4 2 −1 3 D4 = a,b | a = b = e,b ab = a , 4 2 2 −1 3 Q2 = a,b | a = e,b = a ,b ab = a , see pages 58 and 59. The first group, which is isomorphic to the dihedral group of the square, has five elements of order 2. The second is the quaternion group (sometimes called the first dicyclic group), it contains one element of order 1, one element of order 2, and six elements of order 4. As these groups have different numbers of elements of order 2 they are not isomorphic, and so there are exactly two isomorphism classes of non-Abelian groups of order 8. The group Q2 also has one property that it shares with very few others: it is not Abelian and all of its subgroups are normal (groups with this property are called Hamiltonian), see Problem 7.13. Historical note In the middle of the nineteenth century, the Irish mathematician W.R. Hamilton (1805Ð1865) was looking for fields which extend the complex num- bers, he discovered the quaternions which have some remarkable properties but do not (quite) form a field. They can be defined in a similar manner to the complex numbers (that is, as a vector space, now of dimension 4, over R with a vector mul- tiplication) except that the number i is replaced by three entities i, j and k which satisfy i2 = j 2 = k2 =−1 and ij = k =−ji, and a general quaternion is an entity of the form x +iy +zj +tk where x,y,z,t ∈ R. All field axioms are satisfied except one, commutativity of multiplication fails as can be seen above (a system of this type is known as a division algebra). The oc- tonions which are sometimes called Cayley numbers extend this construction to di- mension 8, but in this case both commutativity and associativity fail; some brief details are given in Rose (2002), page 176. The ring of integral quaternions has a similar definition to that for the quaternions given above except now x,y,z,t ∈ Z. The quaternion group Q2 forms the group of units (divisors of 1) of this ring when we map a → i and b → j, for then ab → ij where ij = k. 6.2 Sylow Theory We have seen earlier that the converse of Lagrange’s Theorem is false—if m | o(G) it does not follow that G has a subgroup of order m. For instance, A5 has no sub- group of order 15, see page 101. But there is a partial converse if we restrict m to be a prime power—if pr | o(G), then G does have a subgroup of order pr , and this is the starting point for the Sylow theory. As for Cauchy’s Theorem, we shall give a number of proofs of this important (existence) result. The first uses a new action and is due to Wielandt, and the second uses the Class Equations but as with the second 120 6 p-Groups and Sylow Theory proof of Cauchy’s result, it has the disadvantage that the Abelian case must be dealt with separately. The third proof uses some matrix theory. We begin with the main existence theorem first proved by the Norwegian math- ematician Ludwig Sylow (1832Ð1918) in 1872. Theorem 6.7 (Sylow Theorem, Part 1) If G is finite group with order pr m, where p is prime and p m, then G contains a subgroup of order pr . Proof There is nothing to prove if r = 0, so we may assume that r>0. The following fact concerning binomial coefficients will be used: pr m The binomial coefficient is not divisible by p. (6.2) pr pr m r (The numerator of pr has the following factors (p in total): − − pr m pr m − 1 ··· p pr 1m − 1 pr m − (p + 1) ··· p2 pr 2m − 1 − − ··· p pr 1m − pr 1 − 1 ··· pr m − pr − 1 and when this expression is divided by the denominator of the binomial coef- ficient, that is (pr )!, all factors of the form pi are removed by cancellation.) We define a new action. Let X denote the set of all unordered subsets with r no repetitions of the underlying set of G which contain exactly p elements. pr m As there are pr ways of choosing these subsets, we have by (6.2) p o(X ). (6.3) The action of G on X is defined by right multiplication: If X1 ∈ X then X1\g ={xg : x ∈ X1} for g ∈ G. r Using cancellation we have o(X1\g) = o(X1) = p , and so X1\g ∈ X ,for all g ∈ G. The action axioms (5.1) follow. Let X1, X2,...denote the orbits of this action, as X is a disjoint union of its orbits, we see by (6.3) that there is at least one orbit Xj , say, with the property p o(Xj ). (6.4) ∈ X ∈ ={ }∈X \ −1 ={ } Let Y j where e Y (if Y y1,... j , then Y y1 e,... is in the same orbit as Y by definition), note that o(Y) = pr .NowJ is defined by r J = stabG(Y ). By Lemma 5.6, J ≤ G, hence if we can show that o(J) = p , the result will follow. We do this by proving both pr ≤ o(J), and pr ≥ o(J). First, by the OrbitÐStabiliser Theorem (Theorem 5.7), we have r o(Xj ) = G : stabG(Y ) =[G : J ]=p m/o(J ), 6.2 Sylow Theory 121 using Lagrange’s Theorem for the final equation. Now the last entity in this sequence of equations is an integer not divisible by p (by (6.4)), and so pr | o(J), which implies pr ≤ o(J). (6.5) For the reverse, note that J ⊆ YJ = Y (as e ∈ Y and J is the stabiliser of Y ). Hence o(J) ≤ o(Y) = pr , which with (6.5) proves the theorem. 3 Example We illustrate the above construction with the group D3 a,b | a = b2 = (ab)2 = e which has order 6, and we look for a subgroup of order 3. Here r = = 6 = p 3, m 2 and so the binomial coefficient 3 20, and 3 20. Hence in this example X has 20 elements—the underlying set of D3 has 20 unordered 3-element subsets, and using the action defined above these split into the four orbits given by: {e,a,a2}, {b,ab,a2b}; {e,a,b}, {e,a2,ab}, {e,b,ab}, {a,a2,a2b}, {a,b,a2b}, {a2,ab,a2b}; {e,a,ab}, {e,a2,a2b}, {e,ab,a2b}, {a,a2,b}, {a,b,ab}, {a2,b,a2b}; {e,a,a2b}, {e,a2,b}, {e,b,a2b}, {a,a2,ab}, {a,ab,a2b}, {a2,b,ab}. The first orbit has an order not divisible by 3, and we can take its triple which contains e, that is, {e,a,a2} for Y and J . Now this stabiliser provides the subgroup of order 3 that we were looking for. Reader, find a subgroup of order 2. Second proof of Theorem 6.7 We use induction on o(G) = pr m, there is nothing to prove if o(G) = 1. We treat the cases of G non-Abelian, and G Abelian, separately. The first of these has two subcases. Subcase 1.1 Z(G) < G and p | o(Z(G)). By Cauchy’s Theorem (Theorem 6.2) and as p divides o(Z(G)), Z(G) contains an element of order p, and so it contains a subgroup H of order p.By Problem 2.14(ii), H G, hence we can form the factor group G/H which has order pr−1m. By the inductive hypothesis, this factor group has a subgroup J/H of order pr−1, and the Correspondence Theorem now shows that J